SPECIES AND HYPERPLANE ARRANGEMENTS
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Jose Dario Bastidas Olaya
August 2021
© 2021 Jose Dario Bastidas Olaya
ALL RIGHTS RESERVED
SPECIES AND HYPERPLANE ARRANGEMENTS
Jose Dario Bastidas Olaya, Ph.D.
Cornell University 2021
This dissertation has two leading characters: Hopf monoids in the category of
species and the Tits algebra of a real hyperplane arrangement. The relation between these two comes from the work of Aguiar and Mahajan (2013), who showed
that a (co)commutative Hopf monoid gives rise to a family of (left)right-modules
over the Tits algebra of the braid arrangement in all dimensions. One goal of this
thesis is to explore the representation theory of the Tits algebra of arbitrary affine
arrangements to extend what is known in the case of linear arrangements and to
give an insight into some unanswered questions in the field of Hopf monoids.
In the first part, we extend the study of characteristic elements of a hyperplane
arrangement from the linear to the affine case. We present the basic properties of
these elements and apply them to derive numerous results about the characteristic
polynomial of an arrangement, from Zaslavsky’s formulas to more recent results of
Kung and of Klivans and Swartz. We construct several examples of characteristic
elements, including one in terms of intrinsic volumes of faces of the arrangement.
In the second part, we study deformations A of a linear arrangement A0 and
endow the Tits algebra of A with a bimodule structure over the algebra of A0 . The
left module structure sheds some light on the study of exponential sequences of
arrangements, in the sense of Stanley. In particular, we construct the Hopf monoid
of faces associated with such a sequence and use characteristic elements to deduce
formulas for certain bivariate polynomial invariants of these arrangements.
In the third part, we endow the polytope subalgebra of deformations of a zono-
tope with the structure of a module over the Tits algebra of the corresponding
hyperplane arrangement. We study algebraic invariants of this module and find
relations between statistics on (signed) permutations and the module structure in
the case of (type B) generalized permutahedra. In type B, the module structure
surprisingly reveals that any family of generators (via signed Minkowski sums) for
generalized permutahedra of type B will contain at least 2d−1 full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally,
we prove that the relations defining the polytope algebra are compatible with the
Hopf monoid structure of generalized permutahedra, and explain the relationship
between the antipode formula of this Hopf monoid and inversion in the polytope
algebra.
In the last chapter, we introduce a novel definition of type B Hopf monoids.
Unlike standard Hopf monoids and the Hopf monoids in H-species of Bergeron
and Choquette (2009), our notion involves a pair of species (one of type A with an
involution and one of type B) and a (co)module structure of one over the other.
This closer represents the algebraic structure that arises from the Tits algebra of
the type B Coxeter arrangement. We study some general constructions like the
substitution product of type B objects and the free (commutative) monoid over a
positive type B object. We conclude by endowing the type B object of generalized
permutahedra and the type B object of symplectic matroids with the structure of
a type B Hopf monoid.
BIOGRAPHICAL SKETCH
Jose Dario Bastidas Olaya was born to Jose Luis Bastidas and Martha Irene Olaya
in Pasto, Nariño, Colombia on July, 1991. He attended school at Instituto Champagnat in his hometown before moving to Bogotá to study at Universidad de Los
Andes. There, he completed a Bachelor of Science and a Master of Science in Mathematics under the supervision of Professor Mauricio Velasco. He began graduate
school at Cornell University in 2015, and in 2019 he obtained a Special Masters in
Computer Science. He completed his dissertation in 2021 under the supervision of
Professor Marcelo Aguiar.
iii
Para mis viejos y abuelos.
iv
ACKNOWLEDGEMENTS
First and foremost, I want to express my gratitude to my advisor Marcelo
Aguiar for his endless support during my years at Cornell. I highly appreciate the
many hours Marcelo spent discussing mathematics, asking questions, suggesting
projects, and promoting my academic, professional, and personal growth. I could
not have asked for a better mentor.
I thank Mike Stillman and Ed Swartz for the enthusiasm and dedication shown
in the classroom and for taking the time to be on my committee. Special thanks
to Jim Utz and Melissa Totman for helping me navigate the paperwork and deadlines of graduate school. To all the staff and fellow students of the Mathematics
Department at Cornell, thanks for making graduate school a fantastic experience.
I also want to thank Mauricio Velasco for guiding me during the earlier stages
of my career and encouraging me to pursue a graduate program. Mauricio has
an incredible ability to spread his enthusiasm and appreciation for all areas of
mathematics. I hope one day I can do the same with my students. My gratitude
also goes to Federico Ardila, Karola Mészáros, and Vic Reiner for the valuable
conversations and advice during these years.
I am very grateful to all my friends who, in one way or another, either in
Ithaca or afar, helped me to make these the best years of my life. To Ale, for her
companion and unconditional support. To the Colombian community at Cornell,
especially Ana, Andrés, Camila, David, Eugenio, Javi, Juanjo, and Mateo, for
being my family in Ithaca. To Camilo, Jose, Kostas, and Simón for constantly
keeping in touch and for the joyful moments we shared. And to my lifelong friends,
David, Esteban, Koch, Juanca, Julio, Michel, Nicolás, Samuel, and Sebastián, for
always being there when it matters the most.
Last but not least, I want to thank my parents Martha and Jose, my grand-
v
parents Segundo, Stella, Ana, and Roberto, and all my extended family for their
caring, love, and patience. Your encouragement made this dissertation possible.
Gracias.
vi
CONTENTS
Contents
vii
List of Figures
x
1 Introduction
1
2 Preliminaries
2.1 Polyhedral geometry . . . . . . . . . . . . . . .
2.1.1 Polytopes . . . . . . . . . . . . . . . . .
2.1.2 (Unbounded) Polyhedra . . . . . . . . .
2.2 Hyperplane arrangements . . . . . . . . . . . .
2.2.1 Faces, flats and characteristic polynomial
2.2.2 The Tits semigroup . . . . . . . . . . . .
2.2.3 The braid arrangement . . . . . . . . . .
2.2.4 The type B Coxeter arrangement . . . .
2.3 Hopf monoids in the category of Species . . . .
2.3.1 Hopf monoids in a nutshell . . . . . . . .
2.3.2 Series and characters . . . . . . . . . . .
2.3.3 From Hopf monoids to modules . . . . .
2.3.4 Hopf monoids on linearly ordered sets . .
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7
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23
24
25
3 Characteristic elements for real hyperplane arrangements
27
3.1 The Tits algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Characteristic elements . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Definition and basic properties . . . . . . . . . . . . . . . . . 29
3.2.2 Relation to the characteristic polynomial . . . . . . . . . . . 30
3.2.3 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 The fundamental recursion for the characteristic polynomial 32
3.3.2 An identity of Crapo . . . . . . . . . . . . . . . . . . . . . . 33
3.3.3 The characteristic polynomial on a product. An identity of
Kung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Characteristic elements of parameters ±1 . . . . . . . . . . . . . . 35
3.4.1 The unit element . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 The Takeuchi element . . . . . . . . . . . . . . . . . . . . . 36
3.4.3 Application: Zaslavsky’s formulas . . . . . . . . . . . . . . . 36
3.5 The Adams elements . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.1 Braid arrangement . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.2 Type B Coxeter arrangement . . . . . . . . . . . . . . . . . 38
3.5.3 Graphic arrangement . . . . . . . . . . . . . . . . . . . . . . 39
3.5.4 Coordinate arrangement . . . . . . . . . . . . . . . . . . . . 41
3.6 Characteristic elements and valuations . . . . . . . . . . . . . . . . 41
vii
3.7
Main construction: Intrinsic elements . . . . . . . . . . . . . . . . .
3.7.1 Generalization to Cone angles . . . . . . . . . . . . . . . . .
Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . .
43
52
55
4 Deformations of a linear arrangement and exponential sequences
4.1 The action of a central arrangement on its deformations . . . . . . .
4.1.1 The map f0 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 The map i . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Characteristic and intrinsic elements . . . . . . . . . . . . .
4.2 Exponential sequences of arrangements . . . . . . . . . . . . . . . .
4.3 Hopf monoid of flats and faces . . . . . . . . . . . . . . . . . . . . .
4.3.1 Antipode . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Characteristic elements and power series . . . . . . . . . . . . . . .
4.5 Characters and polynomial invariants . . . . . . . . . . . . . . . . .
59
60
62
65
66
67
68
70
71
73
3.8
5 The module of generalized zonotopes modulo McMullen relations 77
5.1 The polytope algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 Definition and structure theorem . . . . . . . . . . . . . . . 81
5.1.2 Subalgebra relative to a fixed polytope . . . . . . . . . . . . 86
5.2 The polytope algebra as a module . . . . . . . . . . . . . . . . . . . 87
5.2.1 The module structure . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 Eulerian idempotents and diagonalization . . . . . . . . . . 90
5.2.3 Simultaneous diagonalization . . . . . . . . . . . . . . . . . 92
5.2.4 First example: the cube and the coordinate arrangement . . 95
5.2.5 The zonotope module of a product of arrangements . . . . . 97
5.3 The module of generalized permutahedra . . . . . . . . . . . . . . . 97
5.3.1 The symmetric group and the Eulerian polynomial . . . . . 98
5.3.2 The module Generalized permutahedra . . . . . . . . . . . . 99
5.3.3 Simultaneous-eigenbasis for the Adams element . . . . . . . 102
5.4 The module of type B generalized permutahedra . . . . . . . . . . . 108
5.4.1 The hyperoctahedral group and the Type B Eulerian polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.4.2 The module of type B Generalized permutahedra . . . . . . 110
5.4.3 Two bases for type B generalized permutahedra . . . . . . . 116
5.5 Hopf monoid structure . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5.1 The McMullen (co)ideal . . . . . . . . . . . . . . . . . . . . 123
5.5.2 Higher monoidal structures . . . . . . . . . . . . . . . . . . 128
6 Type B Hopf monoids
6.1 Type B species . . . . . . . . . . . . . .
6.1.1 Type B generating functions . . .
6.2 Type B bimonoids . . . . . . . . . . . .
6.2.1 Type A species with an involution
6.2.2 The action of Sp on SpB . . . . .
viii
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130
. 130
. 132
. 133
. 133
. 134
6.2.3 Type B objects, monoids, and comonoids . . . . . . .
6.2.4 Type B bimonoids . . . . . . . . . . . . . . . . . . .
6.3 Convolution modules and the antipode . . . . . . . . . . . .
6.3.1 Type A with an involution . . . . . . . . . . . . . . .
6.3.2 Type B . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Type B Hopf monoids and the antipode . . . . . . .
6.4 Characters and polynomial invariants . . . . . . . . . . . . .
6.4.1 Type A with an involution . . . . . . . . . . . . . . .
6.4.2 Type B . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Substitution product of type B objects . . . . . . . . . . . .
6.7 The type B bimonoids of set compositions and set partitions
6.8 The free (commutative) monoid . . . . . . . . . . . . . . . .
6.9 Type B Boolean functions . . . . . . . . . . . . . . . . . . .
6.10 Type B Submodular functions . . . . . . . . . . . . . . . . .
6.11 Type B generalized permutahedra . . . . . . . . . . . . . . .
6.11.1 Isomorphism with (bi)submodular functions . . . . .
6.12 Symplectic matroids . . . . . . . . . . . . . . . . . . . . . .
Bibliography
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179
ix
LIST OF FIGURES
2.1.1 A 2-dimensional polytope P and two of its faces Pv , Pw maximized
in directions v, w, respectively. On the right, the normal fan ΣP
and the normal cones corresponding to the faces Pv , Pw of P . . . .
2.1.2 The normal fan of the polyhedron P = Q + C is the common
refinement of the normal fans of Q and of C. . . . . . . . . . . . .
2.2.1 A 2-dimensional arrangement A together with its poset of faces
(left) and semilattice of flats (right). The Möbius function of the
lattice of flats is shown in red. The characteristic polynomial of A
is χ(A, t) = t2 − 3t + 3. . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Product of faces in two arrangements of rank 2. . . . . . . . . . . .
8
9
11
13
3.7.1 Intrinsic volumes of a 2-dimensional cone in R2 . . . . . . . . . . . .
3.7.2 The k-the dimensional intrinsic volume of P depends only on its
recession cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.1.1 The arrangement A is a deformation of A0 . C = C ·F0 is a chamber
and G is a minimal face of A. . . . . . . . . . . . . . . . . . . . . .
61
5.0.1 Different expressions for the class of the trapezoid above in the
polytope algebra Π(R2 ). . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Vectors v1 , v2 , v3 lie in the same plane. The vectors v1 , v2 are linearly
independent and log[l1 ] log[l2 ] represents the class of a half-open
parallelogram. In contrast, the product log[l1 ] log[l2 ] log[l3 ] is zero. .
5.3.1 The permutahedron in R3 and R4 . . . . . . . . . . . . . . . . . . .
5.3.2 The Adams element αt and some of the associated Eulerian idempotents of the braid arrangement in R3 . . . . . . . . . . . . . . . .
5.4.1 Type B permutahedron in R2 and R3 . . . . . . . . . . . . . . . . .
5.4.2 The type B generators of Theorem 5.4.9 in R2 . In R3 , we only show
the 4 = 23−1 full-dimensional generators. . . . . . . . . . . . . . . .
43
78
84
100
103
110
118
6.6.1 Associativity of the type B substitution product. . . . . . . . . . . 161
x
CHAPTER 1
INTRODUCTION
Species and Hopf monoids
Combinatorial species were originally introduced by Joyal [49] as a tool for studying generating power series from a combinatorial perspective. They provide a
unified framework to study families of combinatorial objects. When a family of
combinatorial objects has natural operations to merge and break structures, the
corresponding species promotes to a Hopf monoid. The theory of Hopf monoids
in the category of species was developed by Aguiar and Mahajan [6, 7] and has
received significant attention in recent years, see [5, 15, 17, 22, 34, 66, 70, 74, 75].
Notably, the Hopf monoid of generalized permutahedra of Aguiar and Ardila [1]
encompasses several Hopf monoids that had previously been studied on a case-bycase analysis.
Roughly speaking, a species h assigns to each finite set I a collection h[I] of
h-structures on I (think of linear orders of I or set partitions of I). A Hopf monoid
is a species h together with product and coproduct maps
µS,T : h[S] × h[T ] → h[I]
(x, y)
and
∆S,T : h[I] → h[S] × h[T ]
7→ x · y
z 7→ (z|S , z/S )
for each finite set I and decomposition I = S t T . These maps need to satisfy
certain (co)unitality, (co)associativity and compatibility axioms.
Hyperplane arrangements and the Tits algebra
In his work on Coxeter complexes Σ, Tits [72, 73] defined the projection of a simplex
G ∈ Σ into another simplex F ∈ Σ. This operation can be extended to a product on
1
the set Σ[A] of faces of an arbitrary real hyperplane arrangement A. Bidigare [26]
studied the induced semigroup structure and its applications to some well-studied
Markov chains; see also [25, 32]. The Tits algebra of a hyperplane arrangement is
the linearization kΣ[A] of the face semigroup. Saliola [65] studied this algebraic
structure and, in particular, constructed a complete system of primitive orthogonal
idempotents for the Tits algebra.
Fix a hyperplane arrangement A in a real vector space. The product of two
faces F and G is the first face you encounter after moving a small positive distance
from an interior point of F to an interior point of G, as illustrated below
F
FC
FG
C = CF
G
Rendezvous: the braid arrangement
The braid arrangement Ad in Rd consists of hyperplanes xi = xj for 1 ≤ i < j ≤ d.
Faces of the braid arrangement are in correspondence with set compositions (i.e.
ordered set partitions) (S1 , . . . , Sk ) of [d] := {1, 2, . . . , d}. More generally, we let AI
denote the braid arrangement in RI for any finite set I.
Take a Hopf monoid h. Then, every face F = (S1 , S2 , . . . , Sk ) of the braid
arrangement AI induces a map
µ
∆
F
F
h[I] −−−−
−→ h[S1 ] × h[S2 ] × · · · × h[Sk ] −−−−
−→ h[I]
obtained by iterating coproducts and products in any meaningful way. A natural
2
question arises:
Does Σ[AI ] 3 F 7−→ µF ◦ ∆F ∈ End(h[I]) induce a module structure?
Aguiar and Mahajan [7] show that the answer is yes whenever h is commutative
(x · y = y · x) or cocommutative (z|S = z/T and z/S = z|T ).
The relation between Hopf monoids and modules over the Tits algebra of the
braid arrangement is a driving idea for the work contained in this dissertation.
Notably, in Chapters 4 and 5, we first study a particular module over the Tits
algebra of an arbitrary linear arrangement, and subsequently construct a Hopf
monoid compatible with the module structure in the case of the braid arrangement.
Furthermore, the algebraic structures surrounding the Tits algebra of Coxeter arrangement of type B inspired the definition of type B Hopf monoid of Chapter 6.
Chapter 3
Simple representations of the Tits algebra kΣ[A] are one-dimensional and indexed
by flats X of the arrangement. Let χX denote the character of the simple module
associated to the flat X. An element w ∈ kΣ[A] of the Tits algebra is characteristic
of parameter t ∈ k if χX (w) = tdim(X) for all flats X of A.
We extend the theory of characteristic elements for real hyperplane arrangements started in [8, Chapter 12] from the linear to the affine case. We present
the basic properties of these elements and apply them to derive numerous results
about the characteristic polynomial of an arrangement, from Zaslavsky’s formulas
to more recent results of Kung.
A main contribution of this chapter is the construction of a characteristic element canonically associated to each arrangement in terms of intrinsic volumes. As
3
a consequence, we recover a result of Klivans and Swartz [50] relating the characteristic polynomial of an arrangement with the intrinsic volumes of its chambers
(maximal faces).
Chapter 4
An affine arrangement A is a deformation of a linear arrangement A0 if the hyperplanes of A are parallel to those of A0 . We endow the Tits algebra kΣ[A] of
a deformation A of A0 with the structure of a bimonoid over kΣ[A0 ], and subsequently deduce that the bimodule structure arises from an inclusion of algebras
kΣ[A0 ] ,→ kΣ[A].
We then specialize to the case of exponential sequences of arrangements in the
sense of Stanley [69]. A sequence of arrangements A = (A1 , A2 , . . . ) is exponential
if Ad is a deformation of the braid arrangement in Rd and for all S ⊆ [d], the
subarrangement of Ad consisting of hyperplanes parallel to xi = xj for i, j ∈ S is
isomorphic to A|S| . The sequences of braid, Shi, Linial, or Catalan arrangements
are examples of exponential sequences.
Given an exponential sequence A , we construct the Hopf monoid of faces of
the sequence ΣA . Elements of ΣA [I] are the faces of the arrangement AI ∼
= A|I|
in RI . The (cocommutative) Hopf monoid structure is given by a (left) action of
kΣ[(An )0 ] on kΣ[An ]. We use characteristic elements and tools from the theory of
Hopf monoids to study bivariate polynomial invariants of the arrangements in A ,
thus generalizing a result of Stanley.
4
Chapter 5
Generalized permutahedra were first introduced by [38] in the context of optimization of submodular functions, and have been of central interest to combinatorialists
because they serve as polyhedral models for several families of combinatorial structures. They have been extensively studied by Postnikov [61], Postnikov, Reiner
and Williams [60], and many others.
Aguiar and Ardila [1] endowed generalized permutahedra with the structure of
a Hopf monoid, and obtained an elegant and simple formula for its antipode. They
observe the similarity between their formula and the formula for inversion in the
polytope algebra of McMullen [55]. One of the contributions of this chapter is to
give an explanation for this similarity.
We first take a more general approach and for every linear hyperplane arrangement A with ambient space V , we define the module of generalized zonotopes of
A. It is at the same time a subalgebra of the polytope algebra Π(V ), and a right
kΣ[A]-module. We study numerical invariants arising from this module structure
and later specialize to the case of the braid arrangement and the type B Coxeter
arrangement. As a surprising consequence, we show that any collection of type B
generalized permutahedra in Rd that affinely spans the corresponding deformation
cone (also called type cone) must contain at least 2d−1 full dimensional polytopes.
In particular, we give two answers to a question of Ardila, Castillo, Eur and Postnikov [14] that asks for a generating family. One attains the theoretical minimum
of 2d−1 full dimensional polytopes, and the other is invariant under the action of
the corresponding Coxeter group.
5
Chapter 6
The past two decades have witnessed different attempts to extend the theory of
species and Hopf monoids to families of objects with groups of symmetries other
than Sn . See [24, 35, 48] for type B (cubical/hyperoctahedral) species, and [64]
for Coxeter species.
We give a novel definition of type B Hopf monoids. In the same spirit as
work of Bergeron and Choquette, the role of finite sets in species is replaced by
finite sets with a fixed-point free involution. However, our theory substantially
differs from that of previous authors in a central idea. Instead of attempting to
define a monoidal structure on the category of type B species, our construction
involves an action of the monoidal category of (standard) species on the category
of Type B species.
We study some general constructions, such as the free (commutative) monoid
over a positive type B object, the substitution product of type B objects, convolution monoids, and a notion of type B antipode. We endow (type B) Boolean
functions, (bi)submodular functions, and (type B) generalized permutahedra with
the structure of a type B Hopf monoid.
Joint work
The contents of Chapters 3 and 4 are joint work with M. Aguiar and S. Mahajan
and Chapter 6 is joint work with M. Aguiar.
6
CHAPTER 2
PRELIMINARIES
2.1
Polyhedral geometry
We review normal cones, normal fans, tangent cones, and recession cones of polyhedra. For more details, see [77].
2.1.1
Polytopes
Let V be a real vector space of dimension d endowed with an inner product h · , · i,
and let 0 ∈ V denote its zero vector. For a polytope P ⊆ V and a vector v ∈ V ,
let Pv denote the face of P maximized in the direction v. That is,
Pv := {p ∈ P : hp, vi ≥ hq, vi for all q ∈ P }.
The (outer) normal cone of a face F of P is the polyhedral cone
NF P := {v ∈ V : F ≤ Pv } = {v ∈ V : F = Pv },
and the normal fan of P is the collection ΣP = {NF P : F ≤ P } of all normal
cones of faces of P . There is a natural order-reversing correspondence between
faces of P and cones in ΣP . For a cone C ∈ ΣP , we let PC ≤ P denote the face
whose normal cone is C. That is PC = Pv for any v ∈ relint(C).
Recall that a fan Σ refines Σ0 if every cone in Σ0 is a union of cones in Σ. We
say that a polytope Q is a deformation of P if ΣP refines ΣQ . The Minkowski
sum of two polytopes P, Q ⊆ V is the polytope P + Q := {p + q : p ∈ P, q ∈ Q}.
7
NPv P
Pv
v
P
w
NPw P
Pw
Figure 2.1.1: A 2-dimensional polytope P and two of its faces Pv , Pw maximized
in directions v, w, respectively. On the right, the normal fan ΣP and the normal
cones corresponding to the faces Pv , Pw of P .
We say that a polytope Q is a Minkowski summand of P if P = Q+Q0 for some
polytope Q0 . The normal fan of P + Q is the common refinement of ΣP and ΣQ ,
see Figure 2.1.2 for an example involving unbounded polyhedra. Hence, ΣP refines
the normal fan of any of its Minkowski summands, and the Minkowski sum of
deformations of P is again a deformation of P .
The f -polynomial of a d-dimensional polytope P is
f (P, z) =
d
X
fi (P )z i ,
i=0
where fi (P ) is the number of i-dimensional faces of P . The h-polynomial of P
is defined by
h(P, z) =
d
X
hi (P )z i = f (P, z − 1).
i=0
The sequences (f0 (P ), . . . , fd (P )) and (h0 (P ), . . . , hd (P )) are the f -vector and hvector of P , respectively. These polynomials behave nicely with respect to the
Cartesian product of polytopes. If P ⊆ V and Q ⊆ V 0 are polytopes, then
f (P × Q, z) = f (P, z)f (Q, z)
h(P × Q, z) = h(P, z)h(Q, z),
where P × Q = {(p, q) ∈ V ⊕ V 0 : p ∈ P, q ∈ Q}.
8
2.1.2
(Unbounded) Polyhedra
A polyhedral cone C ⊆ V is the positive span of a finite collection of vectors in
V . In particular, a polyhedral cone always contains the zero vector 0 ∈ V . In the
present work, cone will always mean polyhedral cone.
A polyhedron P ⊆ V is any set that can be written as a Minkowski sum
P = Q + C, where Q is a polytope and C is a cone. Normal cones and normal
fans are defined just like for (bounded) polytopes.
The recession cone of a polyhedron P is the cone
rec(P ) := {v ∈ V : v + P ⊆ P }.
(2.1.1)
If P = Q + C, where Q is a polytope and C is a cone, then C = rec(P ). On the
other hand, the polytope Q is not completely determined by P (unless P itself is
a polytope, in which case Q = P and C = rec(P ) = {0}).
+
Q
=
C
P
Figure 2.1.2: The normal fan of the polyhedron P = Q + C is the common refinement of the normal fans of Q and of C.
If C is a cone with minimal face O, then NO C is sometimes called the polar
cone of C. In this case, the normal fan ΣC consists precisely of the faces of NO C.
The tangent cone of P at a face F ≤ P , denoted TF P , is the cone
TF P = {v ∈ V : x0 + tv ∈ P for small enough t > 0},
9
(2.1.2)
where x0 is any point in the relative interior of F . It is the polar cone to NF P .
2.2
Hyperplane arrangements
We follow [8] for definitions, see the first chapters for more details.
2.2.1
Faces, flats and characteristic polynomial
Let A be a (real, affine, finite) hyperplane arrangement: a finite collection of affine
hyperplanes in a finite-dimensional real vector space V . A subspace of V obtained
as intersection of hyperplanes in A is called a flat. The set of flats of A is denoted
by L[A], and it forms a graded join-semilattice ordered by inclusion, in particular
it is ranked. The ambient space is the top element of L[A], we denote it by >.
When the arrangement is central, that is when all hyperplanes intersect, L[A]
is a lattice with minimum element equal to the intersection of all hyperplanes in
A, we denote it by ⊥. The rank of the arrangement, denoted by rank(A), is by
definition the rank of the poset L[A].
The characteristic polynomial of A is
X
χ(A, t) :=
µ(Y, >) tdim(Y) ,
(2.2.1)
Y∈L[A]
where µ is the Möbius function of the poset L[A]. It is a monic polynomial of
degree dim(V ).
The arrangement under a flat X is the following collection of hyperplanes
in ambient space X
AX = {H ∩ X : H ∈ A, X 6⊆ H, H ∩ X 6= ∅}.
10
A
H1
H2
H3
Σ[A]
L[A]
>1
µ( · , >)
H1 −1
H2 −1
H3 −1
•1
•1
•1
Figure 2.2.1: A 2-dimensional arrangement A together with its poset of faces (left)
and semilattice of flats (right). The Möbius function of the lattice of flats is shown
in red. The characteristic polynomial of A is χ(A, t) = t2 − 3t + 3.
This is sometimes called the restriction of A to X. The flats of AX are the flats
of A that are contained in X. Hence,
χ(AX , t) =
X
µ(Y, X) tdim(Y) .
(2.2.2)
Y: Y≤X
Similarly, the arrangement over a flat X is
AX = {H : H ∈ A, X ⊆ H}.
Note that AX is a central arrangement in V , since X is precisely the intersection of
all hyperplanes in AX . The flats of AX are the flats of A that contain X. Hence,
χ(AX , t) =
X
µ(Y, >) tdim(Y) .
(2.2.3)
Y: Y≥X
The hyperplanes in A split V into a collection Σ[A] of polyhedra called faces.
Explicitly, the complement in V of the union of hyperplanes in A is the disjoint
union of open subsets of V ; and Σ[A] is the collection of the closures of these regions
together with all their faces. The collection Σ[A] is a poset under containment,
11
its maximal elements are called chambers. If the arrangement A is central, then
Σ[A] contains a unique minimum element O, which we call the central face of
the arrangement, it coincides with the minimum flat.
The arrangement A is essential if the minimal flats are points. In general,
the minimal flats are pairwise parallel affine subspaces of the same dimension.
Intersecting with an orthogonal subspace makes A essential. Faces of A are in
correspondence with faces of the essentialization. The same applies to flats. A
face of A is essentially bounded if the corresponding face of the essentialization
is bounded.
The support of a face F is the smallest flat s(F ) containing it. Equivalently,
it is the affine span of F . The support map
s : Σ[A] → L[A]
(2.2.4)
is surjective and order preserving.
Given two hyperplane arrangements A and A0 in vector spaces V and W ,
respectively, the product arrangement A × A0 is the hyperplane arrangement
in V ⊕ W consisting of hyperplanes
H × W for each H ∈ A and V × H for each H ∈ A0 .
There are natural identifications
Σ[A] × Σ[A0 ] = Σ[A × A0 ] and L[A] × L[A0 ] = L[A × A0 ],
given by
(F, G) ←→ F × G
and
respectively.
12
(X, Y) ←→ X × Y,
(2.2.5)
2.2.2
The Tits semigroup
The set Σ[A] is a semigroup under the Tits product. Informally, the product of
two faces F and G is the first face you encounter after moving a small positive
distance from an interior point of F to an interior point of G, as illustrated below.
Note that F is always a face of F G and F = F G if and only if s(G) ≤ s(F ).
F
G
FC
O
C
FG
F
GF
H = HF
G
Figure 2.2.2: Product of faces in two arrangements of rank 2.
In order to formalize this product, let us first define the sign sequence of a
face. There are two half-spaces H+ and H− associated to each hyperplane H. The
choice of + and − is arbitrary but fixed. For convenience, we also denote H0 = H.
The sign sequence (H (F ))H∈A of a




0




H (F ) = +






−
face F is defined by
if F ⊆ H,
if F ⊆ H+ and F 6⊆ H,
if F ⊆ H− and F 6⊆ H.
Thus, H (F ) = + (resp. H (F ) = −) if and only if relint(F ) is contained in H+ \ H0
(resp. H− \ H0 ). Moreover, the sign sequence uniquely determines F , since:
F =
\
HH (F ) .
H∈A
13
Now we can define the face F G in terms of its sign sequence.



H (F ) if H (F ) 6= 0,
H (F G) =


H (G) otherwise.
For linear arrangements, such as the one on the left in Figure 2.2.2, Σ[A] is a
monoid and the central face O (obtained by intersecting all hyperplanes in A) is
the unit. For affine arrangements, such as the one on the right, this semigroup is
nonunital.
2.2.3
The braid arrangement
The braid arrangement Ad in Rd consists of the diagonal hyperplanes xi = xj
for 1 ≤ i < j ≤ d. Its central face is the line perpendicular to the hyperplane x1 +
· · · + xd = 0. Intersecting Ad with this hyperplane and a sphere around the
origin we obtain the Coxeter complex of type Ad−1 . The pictures below show the
cases d = 3 and 4.
Flats and faces of Ad are in one-to-one correspondence with set partitions and
set compositions of [d] := {1, 2, . . . , d}, respectively. We proceed to review this
correspondence.
A weak set partition of a finite set I is a collection X = {S1 , . . . , Sk } of
pairwise disjoint subsets Si ⊆ I such that I = S1 ∪ · · · ∪ Sk . The subsets Si are the
14
blocks of X. A set partition is a weak set partition with no empty blocks. We
write X ` I to denote that X is a set partition of I. Given a partition X ` [d], the
corresponding flat of Ad is the intersection of the hyperplanes xi = xj for all i, j
that belong to the same block of X, as illustrated in the following example for
d = 8:
x1 = x3 ,
x2 = x5 = x6 = x8
←→
{13, 2568, 4, 7},
where we write 13 to abbreviate the set {1, 3}, 2568 to abbreviate the set {2, 5, 6, 8}
and so on. We use X to denote both a flat of Ad and the corresponding set partition
of [d]. Observe that dim(X) is precisely the number of blocks of X as a partition.
The partial order relation of L[Ad ] becomes the ordering by refinement of set
partitions. That is, X ≤ Y if the set partition X is refined by Y. Recall that X
is refined by Y if every block of X is the union of some blocks in Y. For instance,
{12345678} ≤ {13, 2568, 4, 7} ≤ {1, 28, 3, 4, 56, 7}.
If S ⊆ I is a union of blocks of a partition X ` I, we let X|S ` S denote the
partition of S formed by the blocks of X contained in S. Let X = {S1 , . . . , Sk } ` [d]
be a partition. Then, the choice of a flat Y ≥ X is equivalent to the choice of
partitions Y|Si ` Si for each block of X. With X and Y as above, the Möbius
function of L[Ad ] is determined by
µ(⊥, X) = (−1)k−1 (k − 1)! and µ(X, Y) = µ(⊥, Y|S1 ) . . . µ(⊥, Y|Sk ),
(2.2.6)
where in each factor, ⊥ denotes the minimum partition of Si .
A set composition of I is an ordered set partition F = (S1 , . . . , Sk ). We
write F I to denote that F is a composition of I, and let s(F ) ` I be the
underlying (unordered) set partition. Given a set composition F [d], the corresponding face of Ad is obtained by intersecting the hyperplanes xi = xj whenever
i, j are in the same block of F , and the halfspaces xi ≥ xj whenever the block
15
containing i precedes the block containing j. For example,
x1 = x3 ≥ x4 ≥ x2 = x5 = x6 = x8 ≥ x7
←→
(13, 4, 2568, 7).
We sometimes refer to these as type A set partitions and type A set compositions.
2.2.4
The type B Coxeter arrangement
d
The type B Coxeter arrangement A±
d in R consists of the hyperplanes xi =
xj , xi = −xj for 1 ≤ i < j ≤ d and xk = 0 for 1 ≤ k ≤ d. Its central face is
the trivial cone {0}. The Coxeter complex of type Bd , obtained by intersecting
the arrangement with a sphere around the origin in Rd , is shown below for d = 2
and 3.
Flats and faces of A±
d are in correspondence with signed set partitions and signed
set compositions of [±d] := {−d, −d + 1, . . . , −1, 1, . . . , d − 1, d}, as originally
introduced by Reiner [63].
Let I be a signed set: a finite set with an fixed-point free involution x 7→ x.
For instance, [±d] with involution x = −x. We will denote signed sets with bold
letters I, J , S.
A subset S ⊆ I is said to be involution-exclusive if S ∩ S = ∅, where
S = {x : x ∈ S}. In contrast, S ⊆ I is said to be involution-inclusive
if S = S. In the later case, S is a signed set itself, with involution obtained by
restricting that of I. Given an involution-exclusive subset S ⊆ I, we let ±S be the
16
involution-inclusive set S ∪ S. Sometimes we refer to involution-exclusive subsets
as admissible, and to subsets that are not involution-exclusive as inadmissible.
Note that inadmissible is not the same as involution-inclusive, but any nonempty
involution-inclusive is inadmissible. A maximal admissible subset of I is called
a transversal, its cardinality is necessarily half of the cardinality of I. The
collection of admissible subsets of I is denoted P 0 (I). Observe that P 0 (I) is closed
under intersections but not necessarily under unions.
A signed set partition of I is a weak set partition of the form X =
{S0 , S1 , S1 , . . . , Sk , Sk }, where S0 is involution-inclusive and allowed to be empty,
and each Si for i 6= 0 is nonempty and involution-exclusive. We call S0 the zero
block of X. We write X `B I to denote that X is a signed partition of I. Given
a signed partition X `B [±d], the corresponding flat of A±
d is the intersection of
the hyperplanes xi = xj for each i, j in the same block of X, where for k ∈ [d],
we let xk denote −xk . In particular, if k ∈ [d] is in the zero block of X, the corresponding flat lies in the hyperplane xk = 0. For instance, consider the following
two examples for d = 7:
x1 = x3 ,
x1 = x3 = 0,
x2 = −x4 = x5 ,
x2 = −x4 = x5 ,
x6 = x7
x6 = x7
←→
{∅, 13, 1̄3̄, 24̄5, 2̄45̄, 67, 6̄7̄},
←→
{11̄33̄, 24̄5, 2̄45̄, 67, 6̄7̄}.
The zero block in the first partition is empty since the corresponding flat is not
contained in any coordinate hyperplane. We use X to denote both a flat of A±
d and
the corresponding signed partition of [±d]. Observe that the number of nonzero
blocks of X is 2 dim(X). The partial order relation of L[A±
d ] becomes the ordering by refinement of signed partitions. For instance, {11̄33̄, 24̄5, 2̄45̄, 67, 6̄7̄} ≤
{∅, 13, 1̄3̄, 24̄5, 2̄45̄, 67, 6̄7̄} ≤ {∅, 13, 1̄3̄, 25, 2̄5̄, 4, 4̄, 67, 6̄7̄}.
If S is an involution-inclusive union of blocks of X `B I, we let X|S `B S
denote the corresponding signed partition. If, on the other hand, S is an involution17
exclusive union of blocks of X `B I, we let X|S ` S denote the corresponding type
A partition.
Let X = {S0 , S1 , S1 , . . . , Sk , Sk }. A choice of Y ≥ X is equivalent to the choice
of a signed partition Y|S0 `B S0 of the zero block and of type A partitions Y|Si ` Si
for i = 1, . . . , k. Note that in this case, Y|Si is automatically determined by Y|Si .
With X and Y as above, the Möbius function of L[A±
d ] is determined by
µ(⊥, X) = (−1)k (2k − 1)!! and µ(X, Y) = µ(⊥, Y|S0 )µ(⊥, Y|S1 ) . . . µ(⊥, Y|Sk ),
(2.2.7)
where (2k − 1)!! denotes the double factorial (2k − 1)!! := (2k − 1)(2k − 3) . . . 1,
and in each factor ⊥ denotes the minimum (signed) partition of (S0 ) Si .
A signed composition is an ordered signed set partition with the property
that Si precedes Sj if and only if Sj precedes Si . The following examples illustrate
the identification between signed compositions of ±[d] and faces of A±
d:
x6 = x7 > −x2 = x4 = −x5 > x1 = x3 > 0
←→
(67, 2̄45̄, 13, ∅, 1̄3̄, 24̄5, 6̄7̄)
x6 = x7 > −x2 = x4 = −x5 > x1 = x3 = 0
←→
(67, 2̄45̄, 11̄33̄, 24̄5, 6̄7̄)
Note that we can alternatively describe the first face with the inequalities
0 > −x1 = −x3 > x2 = −x4 = x5 > −x6 = −x7 .
2.3
Hopf monoids in the category of Species
A comprehensive introduction to the theory of species can be found in the work by
Bergeron, Labelle, and Leroux [23]. The category of species possesses more than
18
one monoidal structure. Of central interest for the present work are the Cauchy
and Hadamard product. Aguiar and Mahajan [6, 7] have explored these structures
extensively, and have exploited this rich algebraic structure to obtain outstanding
combinatorial results.
2.3.1
Hopf monoids in a nutshell
Let set× denote the category of finite sets with bijections as morphisms, and Set
the category of sets and arbitrary set functions. The category of set species Sp is
the functor category [set× , Set]. Explicitly, a set species p consists of the following
data:
1. For each finite set I, a set p[I] of p-structures.
2. For each bijection σ : I → J, a function p[σ] : p[I] → p[J]. These functions
satisfy
p[σ ◦ τ ] = p[σ] ◦ p[τ ]
and
p[Id] = Id .
In particular, they are bijections. We call these maps relabeling maps.
A species p is said to be connected if p[∅] consists of exactly one element. Unless
otherwise stated, we assume all species to be connected and use to denote the
only p-structure on the empty set. A morphism of species f : p → q is a natural
transformation; that is, a collection of functions
fI : p[I] → q[I],
one for each finite set I, that commute with bijections. Namely, fJ ◦p[σ] = q[σ]◦fI
for any bijection σ : I → J.
19
The category of species Sp is a braided monoidal category under the Cauchy
product of species, defined by
(p · q)[I] =
a
p[S] × q[T ].
(2.3.1)
I=StT
A monoid in (Sp, ·) is a species m together with a morphism µ : m · m → m
satisfying certain unitality and associativity axioms. Breaking down this definition,
a monoid is a species m with a collection of maps
µS,T : m[S] × m[T ] → m[I]
(x, y)
7→ x · y,
one for each finite set I and decomposition I = S t T . These maps satisfy
·x=x·=x
(x · y) · z = x · (y · z),
and
for all decompositions I = R t S t T , and structures x ∈ m[R], y ∈ m[S], z ∈ m[T ].
A monoid m is commutative if x · y = y · x for all I = S t T , and x ∈ m[S],
y ∈ m[T ]. A monoid morphism f : m → m0 is a morphism of species that
respects the product: fI (x · y) = fS (x) · fT (y) for all decompositions I = S t T
and structures x ∈ m[S], y ∈ m[T ].
Dually, a comonoid is a species c with a collection of maps
∆S,T : c[I] → c[S] × c[T ]
z 7→ (z|S , z/S ),
one for each finite set I and decomposition I = S t T , satisfying axioms dual to
those of the product. A comonoid c is cocommutative if z|S = z/T and z/S = z|T
for all I = S t T and z ∈ c[I]. Here, z|S is the first component of the coproduct
∆S,T (z), while z/T is the second component of the coproduct ∆T,S (z); both are
structures on S.
20
A Hopf monoid is a species h that is both a monoid and a comonoid and such
that the product and coproduct are compatible, meaning that the coproduct
is a morphism of monoids (or, equivalently, that the product is a morphism of
comonoids). Suppose I = S t T = J t K are two decompositions of the same set I,
and we have structures x ∈ h[J] and y ∈ h[K]. Then, the compatibility axiom
requires that
(x · y)|S = x|A · y|B
and
(x · y)/S = x/A · y/B ,
(2.3.2)
where A = S ∩ J and B = S ∩ K.
Remark 2.3.1. In the general setting, where h is not required to be connected, the
definition above is that of a bimonoid. A Hopf monoid is a bimonoid with an
additional axiom that is automatically satisfied by connected bimonoids. See [6,
Section 2] and Section 6.3.3 for more details.
Example 2.3.2. The exponential species E is the species with exactly one
structure ∗I on each finite set I. It is a bimonoid with the only possible operations:
∗S · ∗T = ∗I and ∆S,T (∗I ) = (∗S , ∗T ) for all decompositions I = S t T .
Example 2.3.3. We consider the species of linear orders L. For any finite set I,
L[I] is the set consisting of all linear orders on I. Given a bijection σ : I → J and
a linear order ` on I, the linear order `0 = L[σ](`) on J is defined by: j <`0 j 0 if
and only if σ −1 (j) <` σ −1 (j 0 ). The species L is a Hopf monoid with the following
operations:
`1 · `2 is the concatenation of `1 and `2 ,
`|S is the restriction of ` to S,
`/S = `|T ,
21
for all I = S t T , ` ∈ L[I], `1 ∈ L[S], and `2 ∈ L[T ]. Representing ` ∈ L[I] with
the list of elements of I ordered according to `, we see that for example
af b · dec = af bdec
∆{a,b,c,d},{e,f } (af bdec) = (abdc, f e).
The compatibility axiom is easily verified; for instance, in the previous example
we have
abdc = ab · dc = af b|{a,b} · dec|{c,d} .
We now shift our attention to the category of vector species Spk which are
obtained by replacing Set with the category Veck (of vector spaces over k with
linear transformations as morphisms) in the definition of Sp. In the present work,
we only consider vector species that are obtained by linearizing a set species.
Given a set species p, the vector species kp is obtained by setting kp[I] to be the
vector space with basis p[I] and by linearly extending the relabeling maps. The
definitions of monoids, comonoids, and bimonoids are obtained by the linearization
of the axioms above.
Let kh be a Hopf monoid. The antipode of kh is the morphism of species
s : kh → kh defined by s∅ = Id, and
sI (x) =
X
(−1)k µS1 ,...,Sk ◦ ∆S1 ,...,Sk (x),
(2.3.3)
I=S1 t···tSk
for I 6= ∅ and x ∈ kh[I]. The sets Si in the sum are nonempty, so they form a
composition of I. We call (2.3.3) the Takeuchi formula for the antipode. The
antipode plays the role of inversion in a Hopf monoid and it is closely related with
reciprocity results for some polynomial invariants that arise from this theory. The
number of terms in (2.3.3) grows extremely fast (∼ n!(log2 (e))n+1 /2) and usually
many cancellations take place. To give a reduced expression for this formula is the
22
content of the antipode problem [6, Section 8.4.2]. In general, the antipode is not
a morphism of monoids nor a morphism of comonoids.
2.3.2
Series and characters
A series of a species kp is a morphism s : kE → kp. That is, a series of kp is a
collection s of elements sI = sI (∗I ) ∈ kp[I], one for each finite set I, such that
kp[σ](sI ) = sJ for each bijection σ : I → J. In particular, the element sI ∈ kp[I]
needs to be invariant under the relabeling maps induced by permutations σ : I → I.
The space of series S (kp) is a k-vector space, with
(c · s)I = csI
(s + t)I = sI + tI
for all s, t ∈ S (kp) and c ∈ k.
We now let kh be a Hopf monoid. In this case, the space S (kh) is an algebra
under the Cauchy product of series s ∗ t, defined by:
X
(s ∗ t)I =
µS,T (sS ⊗ tT ).
(2.3.4)
I=StT
The unit of S (kh) is the series
uI =




if I = ∅,


0 otherwise.
Example 2.3.4. A series of the exponential species kE is a collection s of the
form sI = a|I| ∗I , where a|I| ∈ k only depends on the cardinality of I. Identifying s
P
n
with the power series n≥0 an xn! , one easily verifies that S (kE) ∼
= k[[x]] as vector
spaces. Moreover, the Cauchy product of series (2.3.4) corresponds to the product
of power series, and S (kE) ∼
= k[[x]] is an isomorphism of algebras.
n X
X
n
(s ∗ t)[n] =
a|S| b|T | =
ak bn−k
k
k=0
[n]=StT
23
A series g ∈ S (kh) is said to be group-like if ∆S,T (gI ) = gS ⊗ gT and g∅ = .
If a series g ∈ S (kh) is group-like, then it is invertible under the Cauchy product
(See [7, Section 12.3]), with inverse g −1 determined by
(g −1 )I = sI (gI ).
(2.3.5)
A character ζ of a monoid km is a monoid morphism ζ : km → kE. That is,
a character of km consists of collection of linear maps ζI : km[I] → k such that
ζ∅ () = 1
ζI (x · y) = ζS (x)ζT (y)
ζI (z) = ζJ (m[σ](z)),
for any I = S t T , x ∈ m[S], y ∈ m[T ], z ∈ m[I], and bijection σ : I → J.
If kh is a Hopf monoid, a character ζ of kh defines an algebra morphism
Ψζ : S (kh) → S (kE) ∼
= k[[x]]
as follows. Given a series s of kh, the associated power series is
Ψ(s) = Ψζ (s) =
X
n≥0
2.3.3
ζ[n] (s[n] )
xn
.
n!
(2.3.6)
From Hopf monoids to modules
Let h be a Hopf monoid. The associativity and coassociativity axioms imply that
for any (weak) set composition F = (S1 , . . . , Sk ) I, there are well defined maps
µF : h[S1 ] ⊗ · · · ⊗ h[Sk ] → h[I]
and
∆F : h[I] → h[S1 ] ⊗ · · · ⊗ h[Sk ],
obtained by iterating the product and coproduct maps in any meaningful way. In
particular, the maps µ(I) and ∆(I) are both the identity of h[I].
24
For a finite set I, let AI denote the braid arrangement in RI . Recall that AI
is a central arrangement, and its collection of faces Σ[AI ] forms a monoid under
the Tits product.
Theorem 2.3.5 ([7, Theorem 82]). Let h be a commutative (resp. cocommutative)
Hopf monoid. Then, for every finite set I, h[I] is a right (resp. left) module over
the monoid Σ[AI ]. The action of a face F ∈ Σ[AI ] on a structure x ∈ h[I] is
µF ◦ ∆F (x).
2.3.4
Hopf monoids on linearly ordered sets
Let h be a Hopf monoid in set species. The groupoid of elements el(h) associated
to h is the category whose objects are pairs [I, x] where I is a finite set and x ∈ h[I],
and whose morphisms [I, x] → [J, y] are bijections σ : I → J such that h[σ](x) = y.
An h-species is a functor el(h) → Set. The Hopf monoid structure of h endows
the category of h-species with two monoidal structures. For our purposes, we will
only be interested in the product ? arising from the comonoid structure of h. If p
and q are h-species, then p ? q is defined by
(p ? q)[I, x] =
a
p[S, x|S ] × q[T, x/S ].
(2.3.7)
I=StT
Vector species, Hopf monoids, series and characters are defined for h-species in a
analogous manner. The role of the exponential species E is played by the exponential h-species Eh , that has precisely one structure on each pair [I, x].
We will pay special attention to L-species in Chapter 4. Observe that the
groupoid el(L) is thin: there is at most one morphism from [I, `] to [J, `0 ]. Such a
morphism exists precisely when |I| = |J|, the corresponding bijection is completely
25
determined by ` and `0 . In particular, if kh is an L-species, a series s ∈ S (kh)
corresponds to an arbitrary choice of elements sn ∈ kh[[n], ≤], where ≤ denotes
the usual order of [n].
In view of (2.3.7), a monoid in SpL is an L-species m with maps
µ`S,T : m[S, `|S ] × m[T, `|T ] −→ m[I, `],
one for each finite set I, linear order ` ∈ L[I], and decomposition I = S t T ,
satisfying the usual axioms. A similar observation applies to the collection of
maps ∆`S,T for a comonoid in SpL .
26
CHAPTER 3
CHARACTERISTIC ELEMENTS FOR REAL HYPERPLANE
ARRANGEMENTS
The contents of this chapter are joint work with Aguiar and Mahajan, and have
been partially published in [3]. We further develop the theory of characteristic elements for real hyperplane arrangements started in [8, Chapter 12]. These elements
of the Tits algebra determine the characteristic polynomial of the arrangement and
also determine the characteristic polynomial of the arrangements under each flat.
They are defined by requiring that the simple characters of the Tits algebra evaluate on a characteristic element to powers of a specified parameter.
The Tits algebra is briefly reviewed in Section 3.1. The fact that the characteristic polynomial of an arrangement, which is defined in terms of flats, carries
information about the decomposition of space into faces, originates in work of
Zaslavsky [76], and is at the root of the combinatorial theory of hyperplane arrangements [45, 69]. The Tits algebra provides a natural setting in which this
connection can be further gleaned and developed. Each arrangement possesses
many characteristic elements, and the interest is in constructing particular elements from which specific information about the characteristic polynomial can be
extracted. This chapter illustrates this fact repeatedly.
We review the notion of characteristic elements in Section 3.2, extending the
definitions and main results of [8, Section 12.4] from linear to affine arrangements.
The first applications are given in Section 3.3: we derive the fundamental recursion
for the characteristic polynomial from a basic functoriality property of characteristic elements, and we employ multiplicativity of characteristic elements to derive an
interesting identity due to Kung [52]. Certain characteristic elements of parameters
27
1 and −1 are discussed in Section 3.4, and employed to derive Zaslavsky’s formulas.
Section 3.5 builds characteristic elements for the simplest Coxeter arrangements
in terms of lattice point counting. A main contribution of this chapter is the construction of a characteristic element canonically associated to each arrangement
in terms of intrinsic volumes. This is done in Section 3.7. As an application, we
derive a beautiful result of Klivans and Swartz [50] which relates the coefficients
of the characteristic polynomial to the intrinsic volumes of the chambers.
3.1
The Tits algebra
Let k be a field. The linearization kΣ[A] of the Tits semigroup (Section 2.2.2) is
the Tits algebra of A. See [8, Chapters 1 and 9] for more details. Recall that
the semigroup Σ[A] fails to be unital if the arrangement A is not central. An
interesting fact that we review below (Theorem 3.4.1) is that the Tits algebra is
always unital. We let HF denote the basis element of kΣ[A] associated to the face
F of A.
We view L[A] as a commutative monoid with the join operation ∨ for product.
This makes the support map (2.2.4) a morphism of semigroups. We let HX denote
the basis element of kL[A] associated to the flat X of A, so that HX · HY = HX∨Y .
A result of Solomon [67, Theorem 1] shows that the monoid algebra kL[A]
is split-semisimple. This rests on the fact that the unique complete system of
orthogonal idempotents for kL[A] consists of elements QX uniquely determined by
HX =
X
QY
or equivalently QX =
Y: Y≥X
X
µ(X, Y)HY .
(3.1.1)
Y: Y≥X
In particular, kL[A] is the maximal split-semisimple quotient of kΣ[A] via the sup28
port map and the simple modules of kΣ[A] are indexed by flats. The character χX
of the simple module associated with the flat X evaluated on an element
w=
X
wF HF
(3.1.2)
F
of kΣ[A] yields
X
χX (w) =
wF .
(3.1.3)
F : s(F )≤X
3.2
Characteristic elements
The definitions and results in this section extend those of [8, Section 12.4] to the
setting of affine arrangements.
3.2.1
Definition and basic properties
Let t be a fixed scalar. An element w of the Tits algebra is characteristic of
parameter t if for each flat X
χX (w) = tdim(X) ,
(3.2.1)
with χX (w) as in (3.1.3).
Two characteristic elements of the same parameter take the same value on
all simple modules, and hence differ by a nilpotent element (an element of the
Jacobson radical). The set of characteristic elements of a given parameter is an
affine subspace of the Tits algebra of dimension equal to the number of faces minus
the number of flats.
One-dimensional characters are multiplicative. We deduce the following result.
29
Lemma 3.2.1. If u is a characteristic element of parameter s and v is a characteristic element of parameter t, then uv is a characteristic element of parameter st.
3.2.2
Relation to the characteristic polynomial
The right-hand sides of (2.2.2) and (3.2.1) are related by Möbius inversion, which
implies the following result.
Lemma 3.2.2. An element w of the Tits algebra is characteristic of parameter t
if and only if for every flat X,
X
wF = χ(AX , t).
(3.2.2)
F : s(F )=X
In particular, since the chambers are the faces of top support:
X
wC = χ(A, t),
(3.2.3)
C
with the sum over all chambers C of A.
3.2.3
Functoriality
Let A0 be a subarrangement of A: A0 consists of some of the hyperplanes in A.
There is a morphism of semigroups
f : Σ[A] → Σ[A0 ]
(3.2.4)
which sends a face F of A to the unique face of A0 whose interior contains the
interior of F . This in turn induces a morphism from the Tits algebra of A to that
of A0 : if w is as in (3.1.2), then
f (w) =
X
f (w)G HG , where f (w)G =
G∈Σ[A0 ]
X
F : f (F )=G
30
wF .
This map induces another morphism
f¯ : L[A] → L[A0 ]
(3.2.5)
sending a flat X of A to the minimal flat of A0 containing it. These morphisms
satisfy
s(f (F )) = f¯(s(F ))
(3.2.6)
for all faces F of A.
Lemma 3.2.3. Let w be a characteristic element for A of parameter t, then f (w)
is characteristic for A0 , of the same parameter.
Proof. Let w be a characteristic element of parameter t. Take any flat X of A0 .
Then,
X
s(G)≤X
f (w)G =
X
wF = tdim(X) .
s(F )≤X
Example 3.2.4. Consider the arrangement A0 = AX over a flat X. Fix a face F
with support X. There is a natural bijection between Σ[A]F , the set of faces of A
containing F , and Σ[AX ] [8, Section 1.7.3]. Under this identification, the map
f : Σ[A] → Σ[AX ] = Σ[A]F
is given by
f (G) = F G.
Thus, the interior of the face of AX corresponding to H ∈ Σ[A]F is the union of
the interior of all the faces G of A satisfying F G = H. In the linear case, where
all faces contain the origin, this is precisely the interior of the tangent cone TF H,
which is easily verified from its definition (2.1.2). In the non-linear case, we get
the translation of TF H whose apex (minimal face) contains s(F ).
31
Moreover, for any flat Y of A, we have that
f¯(Y) = X ∨ Y,
since both expressions agree with s(F G), where G is any face with support Y.
3.3
3.3.1
First applications
The fundamental recursion for the characteristic
polynomial
The characteristic polynomial of A may be calculated recursively by removing one
hyperplane at a time. As a first application, we derive a proof of this formula.
Let H be a hyperplane in A and set A0 = A \ {H}. Pick any characteristic element w of parameter t. Applying (3.2.2) to calculate the characteristic polynomial
of AH , and (3.2.3) to calculate that of A, we obtain
χ(A, t) + χ(AH , t) =
X
wC +
X
wF .
The first sum is over all chambers C of A and the second over all faces F of A with
s(F ) = H. By Lemma 3.2.3, we may further employ (3.2.3) to calculate χ(A0 , t) in
terms of coefficients of f (w). We obtain
χ(A0 , t) =
X
f (w)D =
X
wG
the first sum being over all chambers D of A0 and the second over all faces G of A
with f (G) = D for some such D. These faces G are either chambers of A or faces
with support H. Comparing the above expressions, we conclude that
χ(A, t) = χ(A \ {H}, t) − χ(AH , t).
32
(3.3.1)
This derivation of the fundamental recursion is given in [8, Proposition 12.66]
(for linear arrangements). The proof in [69, Lemma 2.2], [58, Theorem 2.56] is
quite different.
3.3.2
An identity of Crapo
We show how to derive the Crapo identity (1.55) in [8], originally proved in [37],
using characteristic elements. We first prove a more general version that was
considered by Kung in [51]. The fundamental recursion above is a particular case
of the following result.
Proposition 3.3.1. For any subarrangement A0 of an arrangement A,
χ(A0 , t) =
X
χ(AX , t),
(3.3.2)
X
where the sum is taken over all flats X of A not contained in any hyperplane of A0 .
Proof. First note that a flat X is not contained in any hyperplane of A0 if and
only if f¯(X) = >. Pick any characteristic element w of parameter t for A. By
Lemma 3.2.3, the element f (w) is characteristic of parameter t for A0 . Applying (3.2.3) to this element, we deduce that
χ(A0 , t) =
X
X
f (w)C =
C
wF .
F : s(f (F ))=>
On the other hand, applying (3.2.2) to w, we obtain
X
f¯(X)=>
χ(AX , t) =
X X
f¯(X)=> F : s(F )=X
wF
=
X
wF .
F : f¯(s(F ))=>
The result follows since the two sums above agree by identity (3.2.6).
33
Let Z be any flat of A and A0 = AZ . From Example 3.2.4 we have that
f¯(X) = X ∨ Z. We deduce the following identity.
Corollary 3.3.2. For any flat Z of an arrangement A,
X
χ(AZ , t) =
χ(AX , t).
(3.3.3)
X: X∨Z=>
3.3.3
The characteristic polynomial on a product.
An
identity of Kung
For the third application we employ Lemma 3.2.1. Pick characteristic elements u
and v of parameters s and t, respectively. Applying (3.2.3) to the characteristic
element uv, we obtain
χ(A, st) =
X
(uv)C =
C
X
uF v G .
F,G: s(F G)=>
The first sum is over chambers C and the second over faces F and G which multiply
to a chamber. This happens precisely when s(F ) ∨ s(G) = s(F G) = >, since s is
a morphism of semigroups. So the previous sum equals
X
X
uF v G .
X,Y: X∨Y=> F : s(F )=X
G: s(G)=Y
Combining the preceding with (2.2.2) we obtain
X
χ(A, st) =
χ(AX , s)χ(AY , t).
X,Y: X∨Y=>
Finally, an application of Corollary 3.3.2 yields
X
χ(A, st) =
χ(AX , s)χ(AX , t).
(3.3.4)
X
This identity is due to Kung [52, Theorem 4]. Kung discusses a couple of proofs,
all quite different from the one above. Kung’s result is for matroids, which covers
the case of linear arrangements. The identity above holds for affine arrangements.
34
3.4
Characteristic elements of parameters ±1
3.4.1
The unit element
When the hyperplanes of A are in general position, the following is [8, Theorem 14.23]. See Section 3.8 for a proof for any affine arrangement.
Theorem 3.4.1. The Tits algebra of an affine arrangement A possesses a unit
element. The unit is
υ=
X
(−1)rank(F ) HF ,
(3.4.1)
F
with F running over the set of essentially bounded faces of A.
When A is linear, the only essentially bounded face is the central face O, and
υ = HO .
The unit element acts as the identity on any module, and hence the onedimensional characters evaluate to 1 on it. This implies the following result.
Proposition 3.4.2. The unit element υ is characteristic of parameter 1.
Here is a direct proof of the proposition. According to (3.1.3), the character
P
value χX (υ) = (−1)rank(F ) is the Euler characteristic of the complex consisting
of the essentially bounded part of X, as illustrated below. The latter is contractible [28, Theorem 4.5.7], and hence the character value is 1.
X
X
1
−1
1
−1
1
35
3.4.2
The Takeuchi element
The Takeuchi element is
τ=
X
(−1)dim(F ) HF ,
(3.4.2)
F
with the sum over all faces F of A.
The following extends [8, Corollary 12.57] to the setting of affine arrangements.
Proposition 3.4.3. The Takeuchi element τ is characteristic of parameter −1.
This time the proof can be brought down to the calculation of the Euler characteristic of a relative pair of cell complexes (B, ∂B), where B is the complex
obtained by dissecting a large ball (containing all the bounded faces) with the
hyperplanes in A.
In the central case, one can more simply work with the reduced Euler characteristic of a sphere, as illustrated below.
−1
1
1
−1
−1
1
1
1
−1
−1
1
1
−1
3.4.3
Application: Zaslavsky’s formulas
All chambers of A are of rank rank(A). Applying (3.2.3) to the unit element υ we
obtain that
(−1)rank(A) χ(A, 1) = (−1)rank(A)
X
Y
36
µ(Y, >)
equals the number of essentially bounded chambers in A.
Let d be the dimension of the ambient space V . Employing the Takeuchi
element τ instead, we obtain that
(−1)d χ(A, −1) =
X
(−1)n−dim(Y) µ(Y, >)
Y
equals the total number of chambers in A. These are Zaslavsky’s formulas [76,
Theorem A, Theorem C, Corollary 2.2], [53, Proposition 8.1].
Remark 3.4.4. The above proof does not differ substantially from Zaslavsky’s.
The core topological argument has been shifted to prove the facts that υ and τ are
characteristic.
3.5
3.5.1
The Adams elements
Braid arrangement
Let Ad be the braid arrangement in Rd (see Section 2.2.3). The Adams element
of type Ad−1 (and parameter t) is defined by
X t αt =
HF ,
dim(F )
F
(3.5.1)
with the sum over the faces F of Ad . For each positive integer k, the binomial
k
coefficient dim(F
counts the number of points in the relative interior of F with
)
coordinates from [k] = {1, . . . , k}. On the other hand, given a flat X, the number
of points in X ∩ [k]d is k dim(X) . Since X splits as the disjoint union of the relatively
open faces F with s(F ) ≤ X, we have that
X k = k dim(X) .
dim(F )
F : s(F )≤X
37
We have shown the following, for which a different proof is given in [8, Lemma
12.78].
Proposition 3.5.1. For any nonzero scalar t, the element αt is characteristic of
parameter t.
There are d! chambers in Ad . As a small application of (3.2.3), we obtain the
well-known expression for the characteristic polynomial of the braid arrangement.
χ(Ad , t) =
X t
C
d
= t(t − 1)(t − 2) · · · (t − (d − 1)).
(3.5.2)
It follows from Lemma 3.2.1 that αs αt is characteristic of parameter st. In fact,
it can be shown that αs αt = αst , see for instance [8, Lemma 12.80].
3.5.2
Type B Coxeter arrangement
Let Bd be the Coxeter arrangement of type B in Rd (see Section 2.2.4). The
Adams element of type Bd (and odd parameter 2t + 1) is defined by
±
α2t+1
=
X
F
t
HF .
dim(F )
(3.5.3)
Proceeding as in Section 3.5.1, but now counting integer points in [−k, k]d ∩ X for
each flat X according to the face in which they lie, one arrives at the following
fact, for which a different proof is given in [8, Proposition 12.89].
±
Proposition 3.5.2. For any scalar t, the element α2t+1
is characteristic of pa-
rameter 2t + 1.
38
There are (2n)!! chambers in A±
d . Employing (3.2.3), one obtains that
χ(A±
d , 2t
t
+ 1) = (2n)!!
,
n
which is equivalent to the familiar expression for the characteristic polynomial of
the type B Coxeter arrangement:
χ(A±
d , t) = (t − 1)(t − 3) · · · (t − (2n − 1)).
(3.5.4)
±
Related elements α2t
are discussed in [8, Section 12.6.3]. These are not charac-
teristic.
3.5.3
Graphic arrangement
Let A(G) be the graphic arrangement associated to a simple graph G. The ambient
space is RI , where I is the vertex set of G, and A(G) contains the hyperplane
xi = xj whenever {i, j} is an edge of G. The arrangement is not essential, its rank
is
rank(A(G)) = |I| − c(G),
where c(G) is the number of connected components of G.
A(G) is a subarrangement of the braid arrangement A in RI . The chromatic
element of G of parameter t is defined by
γt = f (αt ),
where f : kΣ[A] → kΣ[A(G)] is the morphism (3.2.4). Since αt is characteristic
for A, Lemma 3.2.3 implies the following.
Proposition 3.5.3. The element γt is characteristic of parameter t for A(G).
39
Let k be a positive integer. A k-coloring of G is a function x : I → [k], that
is, a point in [k]I . The coloring x is proper whenever vertices i and j are joined
by an edge in G, xi 6= xj . Let p(G, t) denote the chromatic polynomial of G.
Thus, p(G, k) is the number of proper k-colorings of G.
Corollary 3.5.4. The chromatic polynomial of G coincides with the characteristic
polynomial of the associated arrangement. That is,
χ(A(G), t) = p(G, t).
Proof. Applying (3.2.3) to the characteristic element γk = f (αk ), we obtain
χ(A(G), k) =
X
X
αkF =
F :s(f (F ))=>
|[k]I ∩ relint(F )|.
F :s(f (F ))=>
The sums are over faces of the braid arrangement whose image under f is a chamber. A point x lies in the interior of one such face if and only if it does not lie on any
hyperplane of A(G), and this occurs precisely when the coloring x is proper.
Example 3.5.5. Let G be the cycle on 4 vertices. Then A(G) is the smallest
nonsimplicial arrangement. It has 3 types of chambers: triangles, small squares,
and big squares. The small squares are composed of two triangles from the braid
arrangement, and the big squares are composed of four such. The coefficients γkC ,
for the three types of chambers are, respectively,
k
,
4
k
k
2
+
,
4
3
k
k
k
4
+4
+
.
4
3
2
There are 8 triangles, 4 small squares and 2 large ones. It follows that
t
t
t
χ(A(G), t) = 24
+ 12
+2
.
4
3
2
40
3.5.4
Coordinate arrangement
The coordinate arrangement Cd in Rd consists of the coordinate hyperplanes xi = 0
for 1 ≤ i ≤ d. The associated subdivision of the sphere is the Coxeter complex of
T
type Ad1 . The first orthant is i {xi ≥ 0}. For each face F of Cd , let



(t − 1)rank(F ) if F lies in the first orthant,
F
γt =


0
if not.
An argument similar to those in Sections 3.5.1 and 3.5.2, but now counting integer
points in [0, k − 1]d ∩ X, shows that the element
γt =
X
γtF HF
F
is characteristic of parameter t. In this case, only one chamber appears with
nonzero coefficient in γt (the first orthant). We obtain
χ(Cd , t) = (t − 1)d .
Remark 3.5.6. The strategy employed in this section to build characteristic elements draws on ideas of Beck and Zaslavsky in [21], and in fact may be further
developed to study the polynomials introduced in that work.
3.6
Characteristic elements and valuations
The characteristic elements in Section 3.5 are part of a more general family of
examples that arise from a valuation defined on a particular collection of subsets
of the ambient space V . We explain this construction in this section and use it
to define intrinsic elements, a family of characteristic elements defined for any
hyperplane arrangement, in the next section.
41
Let V be a set and C a collection of subsets of V that is closed under finite
intersections and unions. A valuation v on C is a function from C to a commutative
ring R satisfying v(∅) = 0 and
v(C1 ) + v(C2 ) = v(C1 ∩ C2 ) + v(C1 ∪ C2 )
(3.6.1)
for all C1 , C2 ∈ C.
Ehrenborg and Readdy show in [39] that if V is a real vector space and C is the
collection of finite unions of affine subspaces of V , then there is a unique valuation
v : C → Z[t] satisfying
v(A) = tdim(A)
for all affine subspaces A of V .
(3.6.2)
Moreover, for any hyperplane arrangement A = {H1 , . . . , Hk } in V ,
χ(A, t) = v V \
k
[
Hi .
(3.6.3)
i=1
The proof reduces to Möbius inversion.
Fix a hyperplane arrangement A. We can apply formula (3.6.3) as long as we
have a valuation v defined on a collection C of subsets of V containing the boolean
algebra generated by the flats of A and satisfying (3.6.2) on flats of A. One such
collection is the boolean algebra generated by the relative interior of faces of A.
Since the relative interior of different faces of A are disjoint, there is a one-to-one
correspondence between valuations v : C → k and elements w of the Tits algebra
kΣ[A]. Explicitly,
v(C) =
X
wF
←→
wF = v(relint(F )).
(3.6.4)
relint(F )⊆C
The element w is characteristic of parameter t ∈ k if and only if the corresponding
valuation v satisfies (3.6.2) for all flats of A.
42
The characteristic elements of Section 3.5 are constructed from a very simple
kind of valuation: point enumeration. Let P ⊆ V be a finite subset and C any
collection of subsets of V as above. Then, the function v : C → Z defined by
v(C) = |P ∩ C| is a valuation.
Example 3.6.1. Let A be the the braid arrangement in Rd , k a positive integer
and v the valuation given by v(C) = |[k]d ∩ C|. As observed in Section 3.5.1, for
any flat X of A we have |[k]d ∩X| = k dim(X) , so v satisfies (3.6.2). The characteristic
element associated to v under the correspondence (3.6.4) is the Adams element αk .
Remark 3.6.2. As a consequence of the valuation property (3.6.1), the morphism
f : Σ[A] → Σ[A0 ] sends the element associated to a valuation v to the element
associated to the valuation v|C 0 , where C 0 ⊆ C is the boolean algebra generated by
the faces of A0 .
3.7
Main construction: Intrinsic elements
We employ the notion of intrinsic volumes of a convex polyhedral cone C in
Rd [10, Section 2.2]. For each k = 0, . . . , d, let vk (C) be the proportion of volume
of space occupied by points that map to a k-dimensional face of C under the nearest
point projection.
C
α
v2 (C) = α/2π
v1 (C) = 1/2
v0 (C) = 1/2 − α/2π
Figure 3.7.1: Intrinsic volumes of a 2-dimensional cone in R2 .
43
vk (C) is the k-th dimensional intrinsic volume of C. We record the following properties of intrinsic volumes. First and foremost, each intrinsic volume vk is
a valuation on convex cones: it satisfies (3.6.1) whenever C1 ∪ C2 is convex.
If k > dim(C), or if k is smaller than the dimension of the minimal face of C,
then vk (C) = 0. Also, it is clear by definition that
d
X
vk (C) = 1.
(3.7.1)
k=0
The Gauss-Bonnet formula states that if C is not a subspace, then
n
X
(−1)k vk (C) = 0.
(3.7.2)
k=0
If C is a subspace, then
vk (C) =



1
if dim(C) = k,


0
if not.
(3.7.3)
A result by Grünbaum [46] states that
(−1)k vk (C) =
X
(−1)dim(F ) vk (F ).
(3.7.4)
F ≤C
And lastly
0
vk (C × C ) =
k
X
vi (C)vk−i (C 0 ).
(3.7.5)
i=0
We want to extend this notion to convex polyhedra P , using the same definition.
It then turns out that vk (P ) depends only on the recession cone of P (see (2.1.1)
for the definition), and each vk is a valuation on convex polyhedra. To achieve
this, let us first recall a more refined definition in the case of polyhedral cones.
44
rec(P )
P
Figure 3.7.2: The k-the dimensional intrinsic volume of P depends only on its
recession cone.
The solid angle of a cone C is the proportion of the space in the linear span
of C that lies inside of C. That is,
α(C) =
vol(C ∩ B)
,
vol(span(C) ∩ B)
where B is the unit ball centered at the origin and vol is the Lebesgue measure in
span(C). If C is a linear subspace, we have α(C) = 1. We can extend this notion
to an arbitrary polyhedron P by setting
vol(P ∩ Br )
,
r→∞ vol(span(P ) ∩ Br )
α(P ) = lim
where now Br is the ball of radius r and span denotes affine span. It follows that
α(P ) = α(λP ) for any λ > 0. Thus,



α(rec(P )) if dim(P ) = dim(rec(P )),
α(P ) =


0
otherwise.
Moreover, if P and P 0 live in orthogonal subspaces of Rd , then
α(P + P 0 ) = α(P )α(P 0 ).
45
(3.7.6)
Remark 3.7.1. It is worth noting that this notion does not correspond to Beck and
Robins’ definition of solid angle of a polyhedron P at a point p in [20, Chapter
13]. Their definition coincides with the notion of internal angle β defined below.
Given a polyhedron P and a face F ≤ P , the internal and external angles
of P at F are respectively
β(F, P ) = α(TF P ) and γ(F, P ) = α(NF P ).
Note that if C is a cone with apex O, then β(O, C) = α(C).
Let P be a polyhedron and F ≤ P one of its faces. The set of points that project
to the interior of F under the nearest-point map is precisely relint(F ) + NF P .
Define the k-th dimensional intrinsic volume of P as
vk (P ) =
X
α(F )α(NF P ),
(3.7.7)
F ≤P
where the sum is over the k-dimensional faces F of P .
Proposition 3.7.2. For any polyhedron P , vk (P ) = vk (rec(P )). Moreover, vk is
a valuation on polyhedra.
Proof. Let C = rec(P ) and write P = Q + C for some polytope Q. We claim that
for faces G ≤ C and F ≤ P , rec(F ) = G if and only if relint(NF P ) ⊆ relint(NG C).
Indeed, let v ∈ relint(NF P ). Since F = Pv = Qv + Cv , we have Cv = rec(F ).
Thus rec(F ) = G if and only if Cv = G if and only if v ∈ relint(NG C).
We will now use the definition (3.7.7) to compute vk (P ). Observe that:
ˆ For a face F in the sum, if dim(rec(F )) 6= k, then α(F ) = 0 by (3.7.6).
46
ˆ For a k-dimensional face G ≤ C, the only cones in N (P ) that are full-
dimensional inside NG C are of the form NF P for F ≤ P of dimension k with
rec(F ) = G. Since N (P ) refines N (C),
X
relint(NG C) =
relint(NF P ).
dim(F )=k
rec(F )=G
Thus, grouping nonzero terms in (3.7.7) according to rec(F ), we get
vk (P ) =
X
α(F )α(NF P ) =
X
α(G)
G≤C
dim(F )=k
relint(NF P )
X
dim(F )=k
rec(F )=G
=
X
α(G)α(NG C) = vk (C),
G≤C
as we wanted to show. The sums are over k-dimensional faces G of C.
The last statement follows since, whenever P ∪ P 0 is convex,
rec(P ∪ P 0 ) = rec(P ) ∪ rec(P 0 )
rec(P ∩ P 0 ) = rec(P ) ∩ rec(P 0 ).
and
Remark 3.7.3. When P = C is a cone, this definition is equivalent to
vk (C) =
X
β(O, F )γ(F, C)
F ≤C
where O is the minimal face of C and the sum is over the k-dimensional faces F
of C. In this case, all the terms in the sum are nonzero.
We now show that (3.7.4) continues to hold for arbitrary polyhedra.
Lemma 3.7.4. For any polyhedron P ,
(−1)k vk (P ) =
X
(−1)dim(F ) vk (F ).
F ≤P
47
Proof. Let C = rec(P ). Grouping terms according to rec(F ) and using (3.7.4) and
Proposition 3.7.2 we obtain
X
(−1)dim(F ) vk (F ) =
F ≤P
X X
G≤C
(−1)dim(F ) vk (G)
rec(F )=G
X
=
(−1)dim(G) vk (G) = vk (C) = vk (P ).
G≤C
The second equality follows using that, as discussed in the proof of the previous
proposition,
G
relint(NG C) =
relint(NF P )
rec(F )=G
and
X
X
(−1)dim(F ) = (−1)d
rec(F )=G
(−1)dim(NF P )
rec(F )=G
= (−1)d (−1)dim(NG C) = (−1)dim(G) .
Let A be an arrangement in Rd . Each face of A is a polyhedron. We define
the intrinsic element of parameter t for A by
νt =
X
X
(−1)dim(F )
(−1)k vk (F )tk HF .
F
(3.7.8)
k
The following Theorem is a consequence of the discussion in Section 3.6.
Theorem 3.7.5. The element νt is characteristic of parameter t.
Proof. Given that each vk is a valuation, the function v defined for any (closed)
polyhedron P by
v(P ) =
X
vk (P )tk
(3.7.9)
k
extends to a valuation on the Boolean algebra generated by convex polyhedra.
Property (3.7.3) shows that v satisfies condition (3.6.2). Therefore, the element
48
w ∈ kΣ[A] defined by
wF = v(relint(F ))
is characteristic of parameter t. We now verify that νt = w. By inclusion-exclusion,
wC = v(relint(C)) =
X
(−1)dim(C)−dim(F ) v(F )
F ≤C
= (−1)dim(C)
X X
F ≤C
= (−1)dim(C)
(−1)dim(F ) vk (F )tk
(3.7.10)
k
X
(−1)k vk (C)tk = νtC .
k
In the third step we used Lemma 3.7.4.
As an immediate consequence of the theorem, we deduce the following result
of Klivans and Swartz [50, Theorem 5].
Corollary 3.7.6. The coefficient of tk in the characteristic polynomial of A is
(−1)n−k
X
vk (C),
(3.7.11)
C
with the sum over all chambers of A.
Example 3.7.7. The following table shows the intrinsic volumes for the braid
arrangement of rank 3 in its essentialized realization. The ambient space is the
hyperplane x1 + · · · + x4 = 0 in R4 . All faces of the same type are congruent and
hence have the same volumes. The edges of types (2, 1, 1) and (1, 1, 2) have size x
and the edges of type (1, 2, 1) have size y, where
√
1
3
1
1
x=
arccos(
) and y =
arccos( ).
2π
3
2π
3
49
The entries above the main diagonal are 0.
face
center
vertices
v0
v1
v2
y
1
x
1/2
1/2
short edges
1/2 − x
1/2
x
long edges
1/2 − y
1/2
y
1/4
11/24
1/4
triangles
v3
x
1/24
Each circle is composed of four edges of size x and two edges of size y, so
4x + 2y = 1. The arrangement under a circle is combinatorially isomorphic to the
braid arrangement of rank 2. Employing (3.7.11) we obtain that its characteristic
polynomial is
1
1
1
1
4 ( − x) − t + xt2 + 2 ( − y) − t + yt2 = 2 − 3t + t2 .
2
2
2
2
For the characteristic polynomial of the whole arrangement we obtain
1
1 1 11
24 − + t − t2 + t3 = −6 + 11t − 6t2 + t3 .
4 24
4
24
Both calculations agree with (3.5.2) (up to a factor of t from the essentialization).
We turn to general properties of the intrinsic elements. The recession cone
of a face F of A is a linear subspace if and only if F is essentially bounded.
Together with the Gauss-Bonnet formula (3.7.2), this implies that the intrinsic
element of parameter 1 of A is precisely the unit element of the Tits algebra:
ν1 = υ. Formula (3.7.1) implies that the intrinsic element of parameter −1 and
the Takeuchi element coincide: ν−1 = τ .
We now prove a multiplicativity result for intrinsic elements. We first review a
theorem by McMullen that relates the internal and external angles of cones.
50
Theorem 3.7.8 ([54, Theorem 3]). Let C be a cone. Then,



(−1)dim(C) if C is a linear subspace,
X
dim(F )
(−1)
γ(O, F )β(F, C) =


F ≤C
0
otherwise.
(3.7.12)
where O is the minimal face of C.
The Gauss-Bonnet formula (3.7.2) can be expressed in a similar manner, permuting the role of β and γ in the theorem. We will need a version of this result
that works for general polyhedra.
Corollary 3.7.9. Let P be a polyhedron and G ≤ P a fixed face of P . Then,



(−1)dim(P ) if G = P
X
dim(F )
(−1)
α(NG F )α(TF P ) =


F : G≤F ≤P
0
otherwise.
Proof. Consider the tangent cone C = TG P , whose minimal face is O = span(G).
There is a bijection between the faces F of P containing G and the faces of C;
namely, F 7→ TG F . Moreover, NG F = NO (TG F ) and TF P = TTG F C. The result
follows by applying (3.7.12) to C and observing that TG P is a linear subspace if
and only if G = P .
Theorem 3.7.10. For any parameters s and t, νs νt = νst .
Proof. Consider the valuation v as in (3.7.9), so that νtF = v(relint(F )) for all faces
F of A. Following Example 3.2.4, we have that for faces F ≤ P ,
X
G:F G=P
X
νtG =
v(relint(G)) = v(relint(TF P )).
G:F G=P
Fix any face P of the arrangement A. The coefficient of HP in νs νt is
X X
F ≤P
G:F G=P
X
νsF νtG =
νsF v(relint(TF P )).
F ≤P
51
Using the equivalent expressions in (3.7.10), this equals
X X
0
(−1)dim(F ) vk (F )(−s)k (−1)dim(P ) vk0 (TF P )(−t)k
F ≤P
k,k0
0
If we fix k and k 0 , the factor of (−s)k (−t)k in this expression equals
(−1)dim(P )
X
(−1)dim(F ) α(K)α(NK F )α(TF K 0 )α(NK 0 P ),
K≤F ≤K 0 ≤P
where K and K 0 run over faces of dimension k and k 0 , respectively. Taking the
sum over F first, Corollary 3.7.9 implies that the sum is zero unless K = K 0 , in
0
which case k = k 0 and (−s)k (−t)k = (st)k . Thus, the coefficient of HP in νs νt is
(−1)dim(P )
X
(−1)dim(K) α(K)α(NK P )(st)k
K≤P
= (−1)dim(P )
X
P
(−1)k vk (P )(st)k = νst
,
k
and therefore νs νt = νst
Finally, the following proposition is a consequence of the observation in Remark 3.6.2.
Proposition 3.7.11. Let A0 be a subarrangement of A and νt be the intrinsic
element of parameter t of A. Then, f (νt ) is the intrinsic element of parameter t
of A0 , where f is as in (3.2.4).
3.7.1
Generalization to Cone angles
Let Cd be the collection of polyhedral cones in Rd . Backman, Manecke and Sanyal
define in [18] a cone angle in Rd as a map α : Cd → R satisfying the following
three properties.
52
1. The valuation property (3.6.1) whenever C1 ∪ C2 is a cone,
2. α(C) = 0 if dim(C) < d, and
3. α(Rd ) = 1.
Given a cone angle α, the corresponding interior and exterior angles of a polyhedron P at a face F ≤ P are defined by
α̂(F, P ) = α(TF P + L(P )⊥ ) and α̌(F, P ) = α(NF P + L(F )),
respectively, where L(P ) is the linear space parallel to (the affine span of) P . Note
that TF P + L(P )⊥ and NF P + L(F ) are always full-dimensional.
Let A be a central arrangement in Rd , [18, Corollary 4.6 (iii)] shows that
XX
(−1)dim(F ) α̂(O, F )α̌(F, C)tdim(F ) = χ(A, t),
C F ≤C
where the first sum is over the chambers C of A. In fact, this is easily generalized
as follows.
Proposition 3.7.12. Let A be a central arrangement. For any t ∈ k, the element
w ∈ kΣ[A] defined by
wF = (−1)dim(F )
X
(−1)dim(G) α̂(O, G)α̌(−G, −F )tdim(G)
G≤F
is characteristic of parameter t.
We can extend this result to the case of general hyperplane arrangements by
means of the observations in Section 3.6. The result we will prove is the following.
Theorem 3.7.13. Let A be an arbitrary arrangement. Given an angle cone α and
a parameter t ∈ k, the element w ∈ kΣ[A] defined by
wF = (−1)dim(F )
X
(−1)dim(G) α̂(O, G)α̌(−G, − rec(F ))tdim(G)
G≤rec(F )
is characteristic of parameter t.
53
For a polyhedron P , define
X
v(P ) = v(rec(P )) =
α̂(O, F )α̌(F, rec(P ))tdim(F ) ,
F ≤rec(P )
where O = rec(P ) ∩ (− rec(P )) is the minimal face of rec(P ).
Using that rec(P ∪ Q) = rec(P ) ∪ rec(Q) whenever P ∪ Q is a polyhedron, and
rec(P ∩ Q) = rec(P ) ∩ rec(Q) whenever P ∩ Q 6= ∅, we can verify that v defines a
valuation. Moreover, it is clear that v(A) = tdim(A) for any affine subspace. Thus,
the element w ∈ kΣ[A] defined by
wF = v(relint(F ))
is characteristic of parameter t. Note that in the central case this is precisely the
result of Proposition 3.7.12:
v(relint(F )) =
X
(−1)dim(F )−dim(H)
H≤F
=(−1)dim(F )
X
G≤H
X
α̂(O, G)tdim(G)
G≤F
=(−1)dim(F )
α̂(O, G)α̌(G, H)tdim(G)
X
X
(−1)dim(H) α̌(G, H)
H:G≤H≤F
α̂(O, G)tdim(G) (−1)dim(G) α̌(−G, −F )
G≤F
=(−1)dim(F )
X
(−1)dim(G) α̂(O, G)α̌(−G, −F )tdim(G)
G≤F
The second equality follows from [18, Lemma 2.4] with C = NG F + L(G), so that
−C = N−G (−F ) + L(−G).
54
The affine case requires one additional observation.
v(relint(F )) =
X
X
(−1)dim(F )−dim(H)
H≤F
=(−1)dim(F )
α̂(O, G)α̌(G, rec(H))tdim(G)
G≤rec(H)
X
X
(−1)dim(H) α̂(O, G)α̌(G, H 0 )tdim(G)
G≤H 0 ≤rec(F ) rec(H)=H 0
=(−1)dim(F )
0
X
(−1)dim(H ) α̂(O, G)α̌(G, H 0 )tdim(G)
G≤H 0 ≤rec(F )
=(−1)dim(F )
X
(−1)dim(G) α̂(O, G)α̌(−G, − rec(F ))tdim(G)
G≤rec(F )
In the second equality we use that
P
rec(H)=H 0 (−1)
dim(H)
computes the Euler char-
acteristic of the relative pair of cell complexes (X, ∂X), where X = {H : rec(H) ≤
H 0 }.
3.8
Proof of Theorem 3.4.1
The definition of υ in Theorem 3.4.1 only depends on the rank of the faces, not
one their dimension. Thus, in this section we assume without loss of generality
that A is essential.
Let Σb [A] denote the set of bounded faces of A. It forms a cell complex that is
known to be contractible, see [28, Theorem 4.5.7] [69, Chapter 1, Exercise 7]. We
will reduce the proof of Theorem 3.4.1 to the computation of the Euler characteristic of a certain subcomplexes of Σb [A].
Lemma 3.8.1. Let C and D be two different chambers of A. The Euler characteristic of the pair (X, Y ), where
X = {F ∈ Σb [A] : F ≤ C} and
is zero.
55
Y = {F ∈ X : F D 6= C},
Proof. First note that Y is a sub-complex of X: suppose G ∈ Y , F ≤ G but
F ∈
/ Y ; then, GD = GF D = GC = C, contradicting that G ∈ Y .
Consider a large ball B ⊆ V that contains all the bounded faces of A in its
interior. The intersection of B with the faces of A forms the following cell complex
decomposition of B:
Σ[B] := {F : F ∈ Σb [A]} t {F ∩ B, F ∩ ∂B : F ∈ Σ[A] \ Σb [A]}.
Note that Σb [A] ⊆ Σ[B], and for each unbounded face F of A there are two
associated faces F ∩ ∂B l F ∩ B of Σ[B]. Define the following subcomplexes of
Σ[B]:
X 0 = X ∪ {F ∩ B, F ∩ ∂B : F and unbounded face of C}
and
Y 0 = Y ∪ {F ∩ B, F ∩ ∂B : F and unbounded face of C with F D 6= C}.
Since for each unbounded face F of A we have dim(F ∩ B) = dim(F ∩ ∂B) + 1,
the Euler characteristic of (X 0 , Y 0 ) and (X, Y ) agree.
Now, let x be a point in the interior of D ∩ B. Y 0 is precisely the set of
faces of C ∩ B that are visible from x. That is, faces in Y 0 correspond to points
y ∈ C ∩ B such that the segment yx intersects C ∩ B precisely at y. Consider the
map px : C ∩ B → C ∩ B defined by px (y) is the last point inside C ∩ B in the
segment yx. px is well defined since C ∩ B is closed, and it is continuous because
C ∩ B is convex. By the observation above, |Y 0 | is the image of this map, and px
acts as the identity on it. Thus, px : C ∩ B → |Y 0 | is a retraction. Since |Y 0 | is a
retraction of C ∩ B and X 0 is a cell decomposition of C ∩ B, we conclude that
χ(X, Y ) = χ(X 0 , Y 0 ) = 0,
56
as we wanted to show.
Lemma 3.8.2. Let H be a face of A and G < H a proper face. The Euler
characteristic of the pair (T, ∂T ), where
T = {F ∈ Σb [A] : GF ≤ H}
and
∂T = {F ∈ Σb [A] : GF < H},
is zero.
Proof. Define B and Σ[B] as in the proof of Lemma 3.8.1. Similarly, let
T 0 = T ∪ {F ∩ B, F ∩ ∂B : F is an unbounded face such that GF ≤ H}
and
∂T 0 = ∂T ∪ {F ∩ B, F ∩ ∂B : F is an unbounded face such that GF < H}.
Note that the second complex is non-empty precisely because G is a proper face of
H. As before, adding pairs of faces F ∩ B and F ∩ ∂B does not change the Euler
characteristic of the complex, so χ(T, ∂T ) = χ(T 0 , ∂T 0 ).
Finally, note that T 0 is a cell decomposition of TG H ∩ B and ∂T 0 is a cell
decomposition of (∂TG H) ∩ B. Since both of these spaces are contractible, in fact
they are respectively cones over TG H ∩ ∂B and (∂TG H) ∩ ∂B, we have that
χ(T, ∂T ) = χ(T 0 , ∂T 0 ) = 0.
Proof of Theorem 3.4.1. It is enough show that for any face G of A, υHG = HG
and HG υ = HG . We do this by comparing the coefficient of HH on both sides of the
equality, for any face H of A.
First, υHG = HG is equivalent to
X
(−1)rank F =



1 if H = G,


0 otherwise.
: F ∈Σb
F
F G=H
57
(3.8.1)
If H = G, then F G = G = H if and only if F is a face of G. In this case, the
sum computes the Euler characteristic of the bounded complex of G, which, by
arguments similar to those in the previous lemmas, equals χ(G ∩ B) = 1.
If H 6= G, the sum is empty unless s(G) ≤ s(H), so we can assume H is a
chamber C. Let D be the chamber containing G that is furthest from C. Namely,
choose signs so that H (C) = + for all the hyperplanes of A. D is the chamber
containing G such that H (D) = − for all H ≥ s(G). Then, a face F ≤ C
satisfies F G = C if and only if F D = C. Indeed, if F D = C and F G 6= C, then
F G < C and for some hyperplane H (F ) = H (G) = 0; but in that case H (D) = −,
contradicting that F D = C. The result now follows from Lemma 3.8.1.
Now, HG υ = HG is equivalent to
X
(−1)rank F =



1 if H = G,
(3.8.2)


0 otherwise.
: F ∈Σb
F
GF =H
If H = G, then GF = G = H if and only if s(F ) ≤ s(G). Thus, the sum computes
the Euler characteristic of Σb [AX ].
If H 6= G, the sum is trivial unless G is a proper face of H. Fix G < H. Note
that
X
F : F ∈Σb
GF =H
(−1)rank F =
X
(−1)rank F −
F : F ∈Σb
GF ≤H
X
(−1)rank F
F : F ∈Σb
GF <H
computes the Euler characteristic of the pair in Lemma 3.8.2, and is therefore 0.
58
CHAPTER 4
DEFORMATIONS OF A LINEAR ARRANGEMENT AND
EXPONENTIAL SEQUENCES
The contents of this chapter are joint work with Aguiar and Mahajan, and will
appear in an extended version of [3]. An affine arrangement A is a deformation of
a linear arrangement A0 if each hyperplane in A is parallel to some hyperplane in
A0 . Several families of deformations of Coxeter arrangements have been studied by
Athanasiadis, Postnikov, Stanley, Suyama and others. See for instance [16, 62, 71].
In this chapter we consider the Tits algebra of a deformation A as a bimodule
over the Tits algebra of the corresponding linear arrangement A0 . We show that
the left and right action of any element w ∈ Σ[A0 ] on the unit element υ ∈ kΣ[A]
agree, thus concluding the existence of a morphism of algebras i : kΣ[A0 ] → kΣ[A].
We use an explicit description of the map i to prove that the image of the intrinsic
element of parameter t for A0 is the intrinsic element of the same parameter for A.
We then study exponential sequences of arrangements in the sense of Stanley [69]. Given an exponential sequence A , we construct a Hopf monoid of faces
ΣA and a Hopf monoid of flats ΣA of the sequence. The Hopf monoid ΣA is cocommutative, we show that the induced left module structure over the Tits algebra
of the braid arrangement (Section 2.3.4) coincides with the left module structure
above. We use techniques from Hopf monoid theory (in particular series and
characters) to obtain results about polynomial invariants associated with these
sequences, thus extending results of Stanley.
59
4.1
The action of a central arrangement on its deformations
Let A be an affine arrangement. For each hyperplane H ∈ A, let H0 be the
linear hyperplane parallel to H. The linearization of A is the linear arrangement
A0 = {H0 : H ∈ A}. In this situation we also say that A is a deformation of A0 .
Notice that A contains at least one hyperplane parallel to each of the hyperplanes
in A0 .
The set Σ[A] is a bimodule over the monoid Σ[A0 ], as explained next.
Pick F0 ∈ Σ[A0 ] and G ∈ Σ[A], with F0 different from the central face of A0 .
Then a point in the relative interior of F0 determines a direction in the ambient
space (the vector based at the the origin and with vertex at that point). Define
the left and right action of F0 on G as follows:
ˆ F0 · G ∈ Σ[A] is the last face one encounters when walking out of (a point in
the relative interior of) G in the direction of (a point in the relative interior
of) F0 . For the central face O ∈ Σ[A0 ], we set O · G = G.
ˆ G · F0 ∈ Σ[A] is the first face one encounters when walking out of G in the
direction of F0 . For the central face O ∈ Σ[A0 ], we set F · O = F .
See Figure 4.1.1 for an example. Observe that the face F0 · G contains a translate
of F0 . In particular, if F0 is not the central face of A0 , the face F0 ·G is (essentially)
unbounded.
60
A0
A
C
F0 · C
F0
G G · F0
F0 · G
Figure 4.1.1: The arrangement A is a deformation of A0 . C = C · F0 is a chamber
and G is a minimal face of A.
We can define these actions in terms of sign sequences as well. Pick signs so
that for every pair of parallel hyperplanes H and H0 , the translation of H+ and
(H0 )+ to the origin (meaning the origin is in the corresponding hyperplane) agree.
Then, for F0 ∈ Σ[A0 ] and G ∈ Σ[A],



H0 (F0 ) if H0 (F0 ) 6= 0
H (F0 · G) =


H (G)
otherwise,
and
H (G · F0 ) =



H (G)
if H (G) 6= 0,
(4.1.1)
(4.1.2)


H0 (F0 ) otherwise.
With these definitions one easily checks that these products are associative and
unital. We also have the following mixed-associativity axioms:
(F0 · F ) · G0 = F0 · (F · G0 )
and
(F · F0 )G = F (F0 · G).
(4.1.3)
The first one is precisely the bimodule axiom. Similarly, one can check the following
LRB-type properties:
F0 · G · F0 = F0 · G and (G · F0 )G = G · F0 .
(4.1.4)
LRB stands for left regular band : a semigroup S satisfying xx = x and xyx = xy
for all x, y ∈ S.
61
Remark 4.1.1. The right module structure was described by Aguiar and Petersen
in [9, Section 1]. This structure is defined even if A has infinitely many hyperplanes
in each parallelism class (A0 is still required to be finite). However, in that setting
the left action is not well defined (there is no last face).
The bimodule structure above descends to an action of L[A0 ] on L[A] satisfying
s(F0 · G) = s(G · F0 ) = s(F0 ) · s(G) for all F0 ∈ Σ[A0 ] and G ∈ Σ[A]. Explicitly, for
X0 ∈ L[A0 ] and Y ∈ L[A], X0 · Y is the minimum flat of A containing the affine
subspace X0 + Y.
Given a flat X0 of the linear arrangement A0 , we let AX0 be the subarrangement
of A consisting of all hyperplanes that contain a translate of X0 . That is,
AX0 = {H ∈ A : X0 ≤ H0 }.
Unlike the arrangement over a flat (see Section 2.2), this arrangement might not
be central.
4.1.1
The map f0
We define a map
f0 : Σ[A] → Σ[A0 ]
(4.1.5)
as follows. Given F ∈ Σ[A], consider the halfspaces and hyperplanes defining F
and translate them to the origin. Their intersection defines the face f0 (F ) of A0 .
Alternatively, f0 (F ) is the largest face of A0 with a translate contained in F . This
is precisely the recession cone of F , see [77, Proposition 1.12(i)]. In particular, a
face of A is essentially bounded if and only if f0 sends it to the central face of A0 .
62
The map f0 is order-preserving, but is neither a morphism of semigroups, nor
a morphism of Σ[A0 ]-bimodules. However, its linearization f0 : kΣ[A] → kΣ[A0 ]
does send the unit element of kΣ[A] to the unit element of kΣ[A0 ].
The sign sequence of f0 (F ) cannot be immediately read from the sign sequence
of F . However, the following property holds: if H0 (f0 (F )) = + (resp. −), then
H0 = + (resp. −) for all hyperplanes H0 ∈ A parallel to H0 . The converse is not
necessarily true; see the shaded triangle in Figure 4.1.1 for an example. We can
sum this up as follows:
H0 (f0 (F )) 6= 0 ⇒ H (F ) = H0 (f0 (F )).
(4.1.6)
We can similarly define a map
f¯0 : Π[A] → Π[A0 ]
(4.1.7)
sending a flat to its translation to the origin. It is order preserving. Note that
in general f¯0 (s(F )) and s(f0 (F )) do not agree. For example if C is a bounded
chamber, then s(f0 (C)) =⊥ and f¯0 (s(C)) = >.
Property (4.1.6) readily implies that
f0 (F ) · F = F
and F · f0 (F ) = F.
(4.1.8)
Since F0 · G contains a translate of F0 , and f0 (F ) is the largest face of A0 with
a translate contained in F , we see that
F0 ≤ f0 (F0 · G).
(4.1.9)
In particular, if F0 · G = G then F0 ≤ f0 (G). Conversely, suppose that F0 ≤ f0 (G).
Then, using (4.1.8),
F0 · G = F0 · (f0 (G) · G) = F0 f0 (G) · G = f0 (G) · G = G.
63
We have shown the following.
F0 · G = G ⇐⇒ F0 ≤ f0 (G).
(4.1.10)
Now suppose that G · F0 = G. Taking support on both sides we conclude that
s(G) + s(F0 ) is contained in s(G). Therefore, a translate of s(F0 ) is contained in
s(G). Equivalently, s(F0 ) ≤ f¯0 (s(G)). Conversely, suppose s(F0 ) ≤ f¯0 (s(G)). For
any x ∈ relint(F0 ), y ∈ relint(F ), and λ ∈ k, we have y+λx ∈ s(G)+s(F0 ) = s(G).
Moreover, G is full-dimensional inside s(G), so y +λx ∈ relint(F ) for small enough
values of λ > 0. That is, G · F0 = G. We have shown that
G · F0 = G ⇐⇒ s(F0 ) ≤ f¯0 (s(G)).
(4.1.11)
Fix a face F0 of A0 and let X0 = s(F0 ). Let f : Σ[A] → Σ[AX0 ] be the
morphism (3.2.4): it sends a face G of A to the minimal face of the subarrangement
AX0 containing G. That is, H (f (G)) = H (G) for all H ∈ AX0 . Since H0 (F0 ) = 0
for all H ∈ AX0 and H0 (F0 ) 6= 0 for all H ∈ A \ AX0 , it follows that
f (G) = f (G0 ) ⇐⇒ F0 · G = F0 · G0 .
(4.1.12)
Thus, there is a natural bijection
Σ[A]F0 := {F ∈ Σ[A] | f0 (F ) ≥ F0 } −→ Σ[AX0 ]
sending a face F0 · G to f (G). This map is in fact an isomorphism of semigroups.
Under this bijection, relatively bounded faces of AX0 correspond to faces G ∈
Σ[A]F0 such that f0 (G) = F0 .
64
4.1.2
The map i
We have a surjective morphism of semigroups Σ[A] → Σ[A]F0 sending a face G
to F0 · G. Its linearization is a morphism of algebras, and sends the unit element
υ ∈ kΣ[A] (see (3.4.1)) to the unit element of kΣ[A]F0 ∼
= kΣ[AX0 ]. For F ∈ Σ[A]F0
the rank of the corresponding face in Σ[AX0 ] is dim(F ) − dim(F0 ). Therefore,
F0 · υ =
X
(−1)dim(F )−dim(F0 ) HF .
(4.1.13)
F ∈Σ[A]:
f0 (F )=F0
Using (4.1.3) and (4.1.4), we conclude
υ · F0 = (υ · F0 )υ = υ(F0 · υ) = F0 · υ.
We define a map
i : kΣ[A0 ] → kΣ[A]
by i(F0 ) = F0 · υ = υ · F0 . Finally, we can check that bimodule structure is given
by the map i. Indeed,
F0 · G = F0 · (υG) = (F0 · υ)G = i(F0 )G
and
G · F0 = (Gυ) · F0 = G(υ · F0 ) = Gi(F0 ).
Furthermore, we see that i is a morphism of algebras:
i(F0 G0 ) = F0 G0 · υ = F0 · (G0 · υ) = i(F0 )(G0 · υ) = i(F0 )i(G0 ),
and the bimodule structure is precisely the one induced by this algebra extension.
65
4.1.3
Characteristic and intrinsic elements
Proposition 4.1.2. Let v0 ∈ kΣ[A0 ] and w ∈ kΣ[A] be characteristic elements
of parameters s and t, respectively. Then, v0 · w ∈ kΣ[A] is characteristic of
parameter st.
Proof. Take any flat X ∈ L[A]. We have s(F0 · G) ≤ X if and only if s(F ) ≤ f¯0 (X)
and s(G) ≤ X. Then,
X
(v0 · w)F0 ·G =
s(F0 ·G)≤X
X
¯
v0F0 wG = sdim(f0 (X)) tdim(X) = (st)dim(X) .
s(F0 )≤f¯0 (X)
s(G)≤X
Since this occurs for all X ∈ L[A], the result follows.
In particular, letting w = υ and using Proposition 3.4.2, we conclude the
following.
Corollary 4.1.3. If v0 ∈ kΣ[A0 ] is characteristic of parameter s, then so is i(v0 ) ∈
kΣ[A].
Proposition 4.1.4. Let νt0 ∈ kΣ[A0 ] be the intrinsic element for A0 of parameter t. Then, i(νt0 ) = νt , the intrinsic element for A of the same parameter.
Proof. Using (4.1.13) and the definition of the intrinsic element (3.7.8), we have
that the coefficient of HF on i(νt0 ) is
(−1)dim(F )−dim(f0 (F )) (νt0 )f0 (F ) = (−1)dim(F )
X
(−1)k vk (f0 (F ))tk
k
= (−1)dim(F )
X
(−1)k vk (F )tk = (νt )F .
k
In the second step we use that the intrinsic volume of F only depends on f0 (F ),
see Proposition 3.7.2.
66
4.2
Exponential sequences of arrangements
Definition 4.2.1 ([69, Section 5.3]). A sequence A = (A1 , A2 , . . . ) of hyperplane
arrangements is called exponential if
1. Ad is a deformation of the braid arrangement in Rd , and
2. for all S ⊆ [d], the arrangement
(Ad )S := {H ∈ Ad : H is parallel to xi − xj = 0 for some i, j ∈ S}
is combinatorially isomorphic to Ak , where |S| = k.
In particular, the number of hyperplanes in any parallelism class is constant (and
equal to the number of hyperplanes in A2 ).
Well-studied examples of exponential sequences include the Catalan, semiorder,
Linial, and Shi arrangements. The definition of the latter two uses the order of the
elements of [d]. For instance, the Shi arrangement in Rd consists of hyperplanes
xi − xj = 0 and xi − xj = 1 for 1 ≤ i < j ≤ d.
Thus, to define the Shi arrangement in RI for an arbitrary finite set I, we need to
first fix a linear order ` ∈ L[I].
Fix a finite set I and a linear order ` ∈ L[I]. For i, j ∈ I with i <` j, and
α ∈ R, we let Hαi,j denote the hyperplane xi − xj = α in RI , and choose positive
(and negative) half-spaces by setting
(Hαi,j )+ = {x ∈ RI : xi − xj ≥ α}.
If i, j ∈ S ⊆ I, we use Hαi,j to denote both a hyperplane in RS and a hyperplane in
RI , the ambient space will be clear from context.
67
The most natural examples of exponential sequences are obtained by fixing
constants α1 < α2 < · · · < αk and letting Ad consist of hyperplanes
x i − x j = αr
for 1 ≤ i < j ≤ d, 1 ≤ r ≤ k.
Henceforth, we assume our exponential sequences are of this form.
4.3
Hopf monoid of flats and faces
Fix an exponential sequence A = (A1 , A2 , . . . ) as above. Given a finite set I
of cardinality d and a linear order ` = i1 i2 . . . id of I, let A` be the following
hyperplane arrangement:
A` = {Hαia ,ib ⊆ RI : Hαa,b ∈ Ad }.
(4.3.1)
It is a deformation of the braid arrangement in RI .
Definition 4.3.1. The L-species of faces ΣA of the exponential sequence A is
defined by:
ˆ The set ΣA [I, `] is the collection of faces of the arrangement A` , that is
ΣA [I, `] := Σ[A` ].
ˆ Given ` = i1 i2 . . . id ∈ L[I] and `0 = j1 j2 . . . jd ∈ L[J], the relabeling map
ΣA [I, `] → ΣA [J, `0 ] sends a face F ∈ Σ[A` ] to the face F 0 ∈ Σ[A`0 ] with sign
sequence Hαja ,j (F 0 ) = Hαia ,i (F ).
b
b
We proceed to give ΣA the structure of a Hopf monoid in L-species. Fix a
decomposition I = S t T and a linear order ` ∈ L[I].
68
ˆ The product F · G = µ`S,T (F, G) of faces F ∈ ΣA [S, `|S ] and G ∈ ΣA [T, `|T ]
is defined by the sign sequence
Hαi,j (F · G) =



Hα (F ) if i, j ∈ S

i,j






Hα (G) if i, j ∈ T
i,j



+







−
(4.3.2)
if i ∈ S, j ∈ T,
if i ∈ T, j ∈ S.
To verify that this is a valid sign sequence, take any point v ∈ relint(F × G)
P
and consider v + λeS for λ >> 0, where eS = i∈S ei .
ˆ The coproduct ∆`S,T (F ) = (F |S , F |T ) of a face F ∈ ΣA [I, `] is determined by
Hαi,j (F |S ) = Hαi,j (F ). Observe that the product is cocommutative.
Observe that the product arrangement A`|S × A`|T is a subarrangement of A` .
Using the identification Σ[A`|S ] × Σ[A`|T ] = Σ[A`|S × A`|T ] in (2.2.5), the product
and coproduct of ΣA have a very concrete description. Let F ∈ Σ[A`|S × A`|T ],
and take v ∈ relint(F ). Then, µ`S,T (F ) is the only face of A` containing v + λeS
in its interior for λ >> 0. Conversely, if F ∈ Σ[A` ], then ∆`S,T (F ) = f (F ), where
f : Σ[A` ] → Σ[A`|S × A`|T ] is the morphism (3.2.4).
In particular, it follows that
µ`S,T ◦ ∆`S,T (F ) = FS,T · F
(4.3.3)
where FS,T is the face of the braid arrangement in RI corresponding to the composition (S, T ) and the action is that of Section 4.1.
We similarly define the L-species of flats LA , by setting LA [I, `] := L[A` ].
With I = S t T and ` ∈ L[I] as before, the product of flats X ∈ LA [S, `|S ] and
69
Y ∈ LA [T, `|T ] is
X · Y = X ⊕ Y.
(4.3.4)
Analogously, the coproduct is determined by ∆`S,T (X) = (X|S , X|T ), where X|S is
the intersection of the hyperplanes Hαi,j ∈ A`|S such that Hαi,j ∈ A` contains X. The
following result easily follows from the definitions.
Proposition 4.3.2. The collection of support maps s` : ΣA [I, `] → LA [I, `] forms
a morphism of Hopf monoids.
Given a flat X ∈ LA [I, `], we let π(X) ` I be the partition associated to the
flat f¯0 (X) of the braid arrangement in RI , where f0 is the map (4.1.7). That is,
i, j ∈ I are in the same block of π(X) if and only if xi − xj is constant on X. It
follows by the definition of the product of flats (4.3.4) that
π(X · Y) = π(X) t π(Y).
Observe that for i, j ∈ π(X), the hyperplane xi − xj = α containing X might or
might not belong to the arrangement A` . Nevertheless, i, j ∈ I are in the same
α
k−1
block of π(X) if and only if there are hyperplanes Hαi11,i2 , Hαi22,i3 , . . . , Hik−1
,ik ∈ A`
containing X, with i = i1 and j = ik . Therefore, if S ⊆ I is the union of the blocks
in some subpartition π 0 ⊆ π(X), then π(X|S ) = π 0 .
4.3.1
Antipode
We now consider the linearization of these species, kΣA and kLA . Takeuchi’s
formula (2.3.3) and (4.3.3) readily imply that for any w ∈ ΣA [I, `]
s` (w) = τ0 · w,
70
where τ0 is the Takeuchi element of the braid arrangement in RI .
Since τ0 is the intrinsic element of parameter −1, Proposition 4.1.4 implies the
following.
Proposition 4.3.3. The antipode of the Hopf monoid kΣA is given by
s` (w) = τ w,
where τ is the Takeuchi element of A` .
Observe that the antipode formula in the previous proposition is internal to
the algebra kΣ[A` ], and does not involve the Tits algebra of the braid arrangement
at all. The multiplicativity of characteristic elements Lemma 3.2.1 and of intrinsic
elements Theorem 3.7.10 imply the following.
Corollary 4.3.4. The antipode of a characteristic element of parameter t is a
characteristic element of parameter −t. Moreover, the antipode of the intrinsic
element of parameter t is the intrinsic element of parameter −t.
4.4
Characteristic elements and power series
In this section we consider the space of series S (kΣA ) of the Hopf monoid kΣA .
A series corresponds to a sequence (ω1 , ω2 , . . . ), where ωd is an element of the Tits
algebra of Ad .
A series ω ∈ S (kΣ) is a characteristic series of parameter t ∈ k if each
element ω` ∈ kΣA [I, `] is a characteristic element of parameter t.
Proposition 4.4.1. Let ω, $ ∈ S (kΣA ) be characteristic series of parameters s
and t, respectively. Then, ω ∗ $ is a characteristic series of parameter s + t.
71
Proof. To prove that each element (ω ∗ $)` ∈ kΣA [I, `] is characteristic of parameter s + t, we need to show that for all flats X ∈ LA [I, `],
X
(ω ∗ $)G
` =
X
0
ω`|FS $`|FT
(s + t)dim(X) ,
equals
s(G)≤X
where the second sum is over all decompositions I = StT and faces F ∈ ΣA [S, `|S ],
F 0 ∈ ΣA [T, `|T ] such that s(F · F 0 ) ≤ X.
Suppose I = StT , F ∈ ΣA [S, `|S ], and F 0 ∈ ΣA [T, `|T ] are such that s(F ·F 0 ) ≤
X. Then, π(s(F )) t π(s(F 0 )) is coarser than π(X). In particular, S (and T ) is the
union of the blocks in some subpartition π 0 ⊆ π(X). We can then rewrite the
second sum above as
X X
π 0 ⊆π(X) s(F )≤X|S
ω`|FS
X
0
$`|FT
=
s(F 0 )≤X|T
X
0
0
s|π | t|π(X)|−|π | = (s + t)|π(X)| ,
π 0 ⊆π(X)
where S in the first sum is the union of the blocks of π 0 and T = I \ S. Since
|π(X)| = dim(X), the result follows.
The intrinsic series of parameter t is the series ν ∈ S (kΣ) consisting intrinsic
elements of the right parameter; that is, such that ν` ∈ ΣA [I, `] is the in the
intrinsic element of parameter t for A` .
In view of Proposition 4.4.1 and of the multiplicativity of intrinsic elements
(Theorem 3.7.10), one might wonder if the Cauchy of two intrinsic series ν and ν 0
of parameters s and t, respectively, is the intrinsic series of parameter (s + t). This
is not the case, as it can be seen by computing the coefficient (ν ∗ν 0 )F[4] where F is a
long edge as in Example 3.7.7, in the case where each Ad is the braid arrangement
in Rd .
Nevertheless, the intrinsic series satisfies the following interesting property.
72
Proposition 4.4.2. The intrinsic series of parameter t is group-like.
Proof. Using the identification
kΣ[A`|S ] ⊗ kΣ[A`|T ] = kΣ[A`|S × A`|T ],
we verify that both ∆`S,T (ν` ) and ν`|S ⊗ν`|T are the intrinsic element for A`|S ×A`|T of
parameter t. The first statement follows from Proposition 3.7.11, since ∆`S,T (ν` ) =
f (ν` ), where f : Σ[A` ] → Σ[A`|S × A`|T ] is the morphism (3.2.4).
On the other hand
ν`|S ⊗ ν`|T =
X
v(relint(F ))HF ⊗
X
X
0
v(relint(F ))HF 0
F 0 ∈Σ[A`|T ]
F ∈Σ[A`|S ]
=
v(relint(F ))v(relint(F 0 ))HF ⊗ HF 0
(F,F 0 )∈Σ[A`|S ]×Σ[A`|T ]
=
X
v(relint(G))HG
G∈Σ[A`|S ×A`|T ]
is also the intrinsic element for A`|S × A`|T of parameter t. In the last step we used
Proposition 3.7.2, the fact that rec(F × F 0 ) = rec(F ) × rec(F 0 ), and (3.7.5).
An application of (2.3.5) and Proposition 4.3.3 yields the following.
Corollary 4.4.3. The intrinsic series of parameter t is invertible, and its inverse
is the intrinsic series of parameter −t.
4.5
Characters and polynomial invariants
We consider two bivariate polynomial invariants associated to a hyperplane arrangement. First, the Whitney polynomial (or Möbius polynomial) of an ar73
rangement A is defined by
W (A, t, q) =
X
µ(X, Y)tdim(X) q codim(Y) .
X≤Y
On the other hand, the coboundary polynomial of A is
X
χ(A, t, q) =
td−rank(B) (q − 1)|B| ,
central B⊆A
where d is the dimension of the ambient space of A. Observe that d − rank(B) is
the dimension of the flat of A obtained by intersecting the hyperplanes in B. A
result by Ardila [12, Theorem 3.8] gives a formula for χ(A, t, q) in terms of the
Möbius function of L[A]:
χ(A, t, q) =
X
µ(X, Y)tdim(X) q h(Y) ,
X,Y
where h(Y) denotes the number of hyperplanes of A containing Y.
Fix a parameter q ∈ k. We define a character ζ of kΣA by
ζ` (HF ) = q codim(F ) .
Since codim(F ·G) = codim(s(F )⊕s(G)) = codim(s(F ))+codim(s(G)), ζ is indeed
a character. Let w ∈ kΣA [d] be a characteristic element of parameter t. Grouping
faces by their support, we get
X
ζ[d] (w) =
wF q codim(F ) =
F ∈Σ[Ad ]
X
codim(Y)
χ(AY
= W (Ad , t, q).
d , t)q
(4.5.1)
Y∈L[Ad ]
Theorem 4.5.1. Let A = (A1 , A2 , . . . ) be an exponential sequence of arrangements. Then,
xd
1+
W (Ad , t, q) =
d!
d≥1
X
d
d −t
XX
k
d−k x
(−1) fk (Ad )q
1+
,
d!
d≥1 k=1
where fk (Ad ) denotes the number of k-dimensional faces of Ad .
74
(4.5.2)
Proof. Let Ψ : S (kΣA ) → k[[x]] be the algebra morphism associated to the character ζ, as in (2.3.6). If ω ∈ S (kΣA ) is a characteristic series of parameter t,
then (4.5.1) shows that
Ψ(ω) = 1 +
X
W (Ad , t, q)
d≥1
xd
.
d!
In particular, if ω is a characteristic series of parameter −1, Zaslavsky’s formulas
(see Section 3.4.3) imply
Ψ(ω) = 1+
X X
d≥1
codim(Y)
χ(AY
d , −1)q
d
d
X X
k
d−k x
= 1+
.
(−1) fk (Ad )q
d!
d!
d≥1 k=0
xd
Y∈L[Ad ]
The result follows from Proposition 4.4.1 and the fact that Ψ is a morphism of
algebras.
Plugging in q = 0 in (4.5.2), we recover the following result by Stanley.
Corollary 4.5.2 ([69, Theorem 5.17]). Let A = (A1 , A2 , . . . ) be an exponential
sequence of arrangements. Then,
X
χ(Ad , t)
d≥0
xd −t
xd X
=
(−1)d cAd
,
d!
d!
d≥0
where cAd is the number of chambers of the arrangement Ad .
Example 4.5.3. Consider the simplest exponential sequence: for each d, Ad is
the braid arrangement in Rd . Observe that the arrangement AY
d has zero relatively
bounded chambers, unless Y =⊥ is the minimum flat. Thus, W (Ad , 1, q) = q d−1
and plugging in t = 1 in Theorem 4.5.1 yields
1+
X
d≥1
q
d−1 x
d
d!
=
XX
d
d≥0
k=0
k
(−1) fk (Ad )q
d−k
xd −1
d!
.
After a reparametrization, we obtain the following generating function for the
ordered Stirling numbers (k!s(d, k) = fk (Ad ) is the number of compositions of a
75
set of size d into k parts):
X
fk (Ad )q k
d,k≥0
1
xd
=
.
d!
1 − qex + q
We similarly obtain the generating function for the Whitney polynomial of the
braid arrangement
d t
X
t
xd d−1 x
1+
W (Ad , t, q) = 1 +
q
= 1 + 1q (eqx − 1)
d!
d!
d≥1
d≥1
X
Let us now consider a different character. Fix a parameter q ∈ k, and define
the character φ by
φ` (HF ) = q h(F ) ,
where h(F ) = h(s(F )) is the number of hyperplanes of A` containing F . Take
a decomposition I = S t T , a linear order ` ∈ L[I] and faces F ∈ ΣA [S, `|S ],
G ∈ ΣA [T, `|T ]. It readily follows from (4.3.2) that a hyperplane of A` containing
F · G corresponds to either a hyperplane of A`|S containing F or a hyperplane of
A`|T containing G, and thus h(F · G) = h(F ) + h(G). Therefore, φ is a character
of kΣA .
If w ∈ kΣA [d] is a characteristic element of parameter t, then
ζ[d] (w) =
X
wF q h(F ) =
F ∈Σ[Ad ]
X
h(Y)
χ(AY
= χ(Ad , t, q).
d , t)q
(4.5.3)
Y∈L[Ad ]
Following the ideas in the proof of Theorem 4.5.1, we deduce the following.
Theorem 4.5.4 ([12, Theorem 5.2]). Let A = (A1 , A2 , . . . ) be an exponential
sequence of arrangements. Then,
xd X
xd t
.
χ(Ad , t, q) =
χ(Ad , 1, q)
d!
d!
d≥0
d≥0
X
76
CHAPTER 5
THE MODULE OF GENERALIZED ZONOTOPES MODULO
MCMULLEN RELATIONS
Generalized permutahedra serve as a geometric model for many classical (type A)
combinatorial objects and have been extensively studied in resent years. Notably,
Aguiar and Ardila [1] endowed generalized permutahedra with the structure of
a Hopf monoid GP in the category of species and, in so doing, gave a unified
framework to study similar algebraic structures over many different families of
combinatorial objects. At the same time, McMullen’s polytope algebra [55] offers
a different algebraic perspective to study generalized permutahedra.
One of the goals of the present chapter is to investigate the compatibility between both structures. To achieve this goal, we take a more general approach
and study deformations of an arbitrary zonotope. We then specialize our results
to deformations Coxeter permutahedra of type A and type B, revealing remarkable connections with (type B) Eulerian polynomials and statistics over (signed)
permutations. The results in type B allow us to find a family of generalized permutahedra that generates all other deformations via signed Minkowski sums, thus
solving a question posed by Ardila, Castillo, Eur, and Postnikov [14]. Most of the
results presented in this chapter appear in [19].
Let V be a finite-dimensional real vector space. The polytope algebra Π(V ) is
generated by the classes [P ] of polytopes P ⊆ V . These classes satisfy the following
valuation and translation invariance relations: [P ∪ Q] + [P ∩ Q] = [P ] + [Q] and
[P + {t}] = [P ], whenever P ∪ Q is a polytope and for every t ∈ V . The product
structure of Π(V ) is given by the Minkowski sum of polytopes: [P ] · [Q] = [P + Q].
77
h
i ·
=
h
i
=
h
i
+
h
i
−
h
i
Figure 5.0.1: Different expressions for the class of the trapezoid above in the
polytope algebra Π(R2 ).
For a fixed polytope P ⊆ V , Π(P ) denotes the subalgebra of Π(V ) generated
by classes of deformations of P ; see McMullen [56]. We are particularly interested
in the case where P is a zonotope corresponding to a linear hyperplane arrangement A. In this case, let RΣ[A] denote the Tits algebra of A, see Section 5.2.2.
It is linearly generated by the elements HF as F runs through the faces of the
arrangement. The following is a central result of this chapter.
Theorem 5.2.3. Let P and A be as above. The algebra Π(P ) is a right RΣ[A]module under the following action:
[Q] · HF := [Qv ],
where v ∈ relint(F ) and Qv denotes the face of Q maximal in the direction of v.
Moreover, the action of HF is an endomorphism of graded algebras.
In the particular case of the permutahedron and the braid arrangement, the
compatibility between the algebra structure and the action of the Tits algebra is
related to the Hopf monoid structure of Aguiar and Ardila. Similar results for the
Hopf monoid of extended generalized permutahedra were independently obtained
by Ardila and Sanchez in [15].
Theorem 5.5.3. The species Π defined by Π[I] = Π(πI ), where πI ⊆ RI is the
standard permutahedron, is a Hopf monoid quotient of GP.
We embark on further understanding the module structure of Π(Z) when Z is
a zonotope corresponding to a hyperplane arrangement A. The decomposition of
78
Π(Z) =
L
r
Ξr (Z) into its graded components will play an essential role here. Mc-
Mullen characterized Ξr (Z) as the eigenspace of the dilation morphism δλ , defined
by δλ [P ] = [λP ], with eigenvalue λr for any positive λ 6= 1. Theorem 5.2.3 then
implies that each graded component is a RΣ[A]-submodule.
The simple modules over RΣ[A] are one-dimensional and indexed by the flats
of the arrangement [8, Chapter 9]. Given a module M over RΣ[A], the number of
copies of the simple module associated with the flat X that appear as a composition
factor of M is ηX (M ). We propose that studying these algebraic invariants for
the modules Ξr (Z) yields important geometric and combinatorial information of
the deformations of Z. In Sections 5.3 and 5.4, we do this for the algebra of
deformations of the permutahedron and the type B permutahedron, respectively.
The main results in these sections relate the invariants ηX with statistics over
(signed) permutations:
Theorem 5.3.1. For any flat X of the braid arrangement in Rd and r =
0, 1, . . . , d − 1,
ηX (Ξr (πd )) = {σ ∈ Sd : s(σ) = X, exc(σ) = r} .
Theorem 5.4.1. For any flat X of type B Coxeter arrangement in Rd and r =
0, 1, . . . , d,
ηX (Ξr (πdB )) = {σ ∈ Bd : s(σ) = X, excB (σ) = r} .
This surprising relation arises from the remarkable results of McMullen and
Brenti that we explain now. McMullen [56] proved that when P is a simple polytope, just like the (type B) permutahedron, the dimension of the graded components Ξr (P ) are determined by the h-numbers of P . On the other hand, building
on top of work by Björner [27], Brenti proved that the h-polynomial of the Coxeter
79
permutahedron associated with any Coxeter group is the corresponding Eulerian
polynomial [30, Theorem 2.3].
Work of Postnikov [61], and of Ardila, Benedetti, and Doker [13] show that
any generalized permutahedron in Rd can be written as the signed Minkowski
sum of the faces of the standard simplex ∆[d] = Conv{ej : j ∈ [d]}. As a
consequence of Theorem 5.4.1, we show that any such set of generators for type
B generalized permutahedra contains at least 2d−1 full dimensional polytopes (see
Proposition 5.4.8). We manage to obtain a family of generators that attains this
minimum.
Theorem 5.4.9. Every type B generalized permutahedron in Rd can be written
uniquely as a signed Minkowski sum of the simplices ∆S and ∆0S for special
involution-exclusive subsets S ⊆ [±d].
Unlike the standard simplex and its faces in the type A case, this collection of
generators is not invariant under the action of the corresponding Coxeter group.
However, using the set of generators in the previous theorem, we are able to find
a different collection of generators that is invariant under the action of Bd , at the
cost of including twice as many full-dimensional polytopes.
Theorem 5.4.12. Every type B generalized permutahedron in Rd can be written
uniquely as a signed Minkowski sum of the simplices ∆0S for involution-exclusive
subsets S.
This chapter is organized as follows. We review McMullen’s construction in Section 5.1. In Section 5.2.2, we recall the definition of the Tits algebra of a hyperplane
arrangement and some of its representation theoretic properties. Characteristic elements and Eulerian idempotents are also reviewed in this section. In Section 5.2,
80
we begin the study of the polytope algebra of deformations of a zonotope as a
module over the Tits algebra of the corresponding hyperplane arrangement, which
is the central construction for this chapter. We study the polytope algebra of generalized permutahedra in Section 5.3. In particular, we provide a conjectural basis
of simultaneous eigenvectors for the action of the Adams element and positive dilations on the module Π(πd ). Section 5.4 contains the analogous results for type
B. We also give explicit sets of signed Minkowski generators for type B generalized
permutahedra; one shows that the lower bound on the number of full-dimensional
polytopes in such a collection is tight, and the other is invariant under the action of
Bd . In Section 5.5 we prove that the valuation and translation invariance relations
are compatible with the Hopf monoid structure of GP.
5.1
The polytope algebra
We briefly review the construction of McMullen’s polytope algebra [55] and its
main properties. A hands on introduction to this topic can be found in the survey
[33]. The subalgebra relative to a fixed polytope [56] is studied at the end of this
section.
5.1.1
Definition and structure theorem
As an abelian group, the polytope algebra Π(V ) is generated by elements [P ],
one for each polytope P ⊆ V . These generators satisfy the relations
[P ∪ Q] + [P ∩ Q] = [P ] + [Q],
81
(5.1.1)
whenever P , Q and P ∪ Q are polytopes; and
[P + {t}] = [P ],
(5.1.2)
for any polytope P and translation vector t ∈ V . These relations are referred as the
valuation property and the translation invariance property, respectively.
The group Π(V ) is endowed with a commutative product defined on generators by
means of the Minkowski sum
[P ] · [Q] := [P + Q].
It readily follows from (5.1.2) that the class of a point 1 := [{0}] is the unit of Π(V ).
For each scalar λ ∈ R, the dilation morphism δλ : Π(V ) → Π(V ) is defined
on generators by δλ [P ] = [λP ]. Recall that for any subset S ⊆ V and λ ∈ R,
λS := {λv : v ∈ S}. One can easily verify that δλ preserves the valuation (5.1.1)
and translation invariance (5.1.2) relations, and that it defines a morphism of rings.
The main structural result for the polytope algebra is the following.
Theorem 5.1.1 ([55, Theorem 1]). The commutative ring Π(V ) is almost a
graded R-algebra, in the following sense:
i. as an abelian group, Π(V ) admits a direct sum decomposition
Π(V ) = Ξ0 (V ) ⊕ Ξ1 (V ) ⊕ · · · ⊕ Ξd (V );
ii. under multiplication, Ξr (V ) · Ξs (V ) = Ξr+s (V ), with Ξr (V ) = 0 for r > d;
iii. Ξ0 (V ) ∼
= Z, and for r = 1, . . . , d, Ξr (V ) is a real vector space;
iv. the product of elements in
L
r≥1
Ξr (V ) is bilinear.
v. the dilations δλ are algebra endomorphisms, and for r = 0, 1, . . . , d, if x ∈
Ξr (V ) and λ ≥ 0, then δλ x = λr x.
82
We discuss the definition of the graded components Ξr (V ) below. The component Ξ0 (V ) of degree 0 is simply the subring of Π(V ) generated by 1, and
thus Ξ0 (V ) ∼
= Z. Let Z1 be the subgroup of Π(V ) generated by all elements of the
form [P ]−1. Observe that δ0 [P ] = 1 for every polytope P , so Z1 = ker δ0 is an ideal.
As an abelian group, Π(V ) has a direct sum decomposition Π(V ) = Ξ0 (V ) ⊕ Z1 .
Moreover, Z1 is a nil ideal, since for any k-dimensional polytope P ,
([P ] − 1)r = 0 for r > k.
This is [55, Lemma 13]. McMullen also shows that Z1 has the structure of a vector
space (first over Q and then over R). Therefore, we can define inverse maps
log
1 + Z1
Z1
exp
by means of their usual power series. In particular, we can define the log-class of
a k-dimensional polytope P by
log[P ] := log(1 + ([P ] − 1)) =
k
X
(−1)r−1
r=1
r
([P ] − 1)r .
(5.1.3)
Using the exponential map, we recover [P ] from log[P ]:
k
X
1
[P ] =
(log[P ])r .
r!
r=0
(5.1.4)
The log and exp maps satisfy the standard properties of logarithms and exponentials. In particular,
log[P + Q] = log([P ] · [Q]) = log[P ] + log[Q].
(5.1.5)
Example 5.1.2. Let v1 , . . . , vk ∈ V be nonzero vectors and let li denote the
line segment Conv{0, vi }. Then, log[li ] = [li ] − 1. We will see that the product
Qk
i=1 log[li ] ∈ Ξk (V ) is nonzero if and only if the collection {v1 , . . . , vk } is linearly
independent. See Section 5.1.1 for an example with k = 3.
83
Consider the polytope Z = l1 +l1 +· · ·+lk , a Minkowski sum of segments. Using
P
Q
the logarithm property (5.1.5), we get (log[Z])k = ( ki=1 log[li ])k = k! ki=1 log[li ].
Q
The last equality follows since k! log[li ] is the only square-free term in the exP
pansion of ( log[li ])k , and (log[l])2 = ([l] − 1)2 = 0 for any line segment l. Finally, (log[Z])k 6= 0 if and only if k ≤ dim(Z), and Z being the sum of k line segments has dimension at most k, with equality precisely when the vectors v1 , . . . , vk
are linearly independent.
v3
v2
v1
([l1 ] − 1)([l2 ] − 1) =
[l1 ][l2 ]
−
[l1 ]
−
[l2 ]
+
•
=
([l1 ] − 1)([l2 ] − 1)([l3 ] − 1) =
[l1 ][l2 ][l3 ]
−
[l1 ][l2 ]
[l2 ][l3 ]
[l1 ][l3 ]
[l1 ]
[l2 ]
−
+
•
=0
[l3 ]
Figure 5.1.1: Vectors v1 , v2 , v3 lie in the same plane. The vectors v1 , v2 are linearly
independent and log[l1 ] log[l2 ] represents the class of a half-open parallelogram. In
contrast, the product log[l1 ] log[l2 ] log[l3 ] is zero.
For r ≥ 1 let Ξr (V ) be the subgroup (or subspace) of Π(V ) generated by
elements of the form (log[P ])r . The following result implies that the sum Π(V ) =
Ξ0 (V ) + Ξ1 (V ) + · · · + Ξd (V ) is direct, and characterizes each graded component
as the space of eigenvectors for the positive dilations δλ .
Lemma 5.1.3 ([55, Lemma 20]). Let x ∈ Π(V ) and λ > 0, with λ 6= 1. Then,
x ∈ Ξr (V )
if and only if
84
δλ x = λr x.
(5.1.6)
It is clear by the definition of the graded components Ξr (V ) that the class of
a half-open parallelogram in Section 5.1.1 is in Ξ2 (V ). We can also verify this
using (5.1.6) with λ = 2:
!
δ2
=
=
= 22
Convention 5.1.4. As in later work of McMullen [56, 57], we replace Ξ0 (V ) ∼
=Z
with the tensor product Ξ0 (V )R := R ⊗ Ξ0 (V ) ∼
= R to get a genuine graded Ralgebra Π(V )R := Ξ0 (V )R ⊕ Z1 . To simplify notation, we drop the subscript R and
sometimes wirte Ξr and Π instead of Ξr (V ) and Π(V )R .
For an arbitrary vector v ∈ V , we can define a maximization operator P 7→
Pv on the space of all polytopes P ⊆ V . The next result shows that it induces a
well-defined map on Π(V ).
Theorem 5.1.5 ([55, Theorem 7]). The map P 7→ Pv induces an endomorphism
x 7→ xv on Π(V ), defined on generators by
[P ] 7→ [P ]v := [Pv ].
This endomorphism commutes with nonnegative dilations.
In particular, the morphism x 7→ xv restricts to each graded component Ξr .
Lastly, the Euler map x 7→ x∗ is the linear operator defined on generators by
[P ]∗ =
X
(−1)dim(Q) [Q].
(5.1.7)
Q≤P
The sum runs over all nonempty faces Q of P . Up to a sign, the element [P ]∗
corresponds to the class of the interior of P .
85
Theorem 5.1.6 ([55, Theorem 2]). The Euler map is an involutory automorphism
of Π(V ). Moreover, for x ∈ Ξr (V ) and λ < 0,
δλ x = λ r x ∗ .
5.1.2
Subalgebra relative to a fixed polytope
Fix a polytope P ⊆ V . The subalgebra relative to P , denoted Π(P ), is the
subalgebra of Π(V ) generated by classes [Q] of deformations Q of P . It is worth
pointing out that if Q, Q0 are deformations of P such that Q ∪ Q0 is a polytope,
then both Q ∪ Q0 and Q ∩ Q0 are deformations of P . This follows since, in this case,
Q ∪ Q0 + Q ∩ Q0 = Q + Q0 and Minkowski summands of a deformation of P are
again deformations of P . Thus, the valuation property (5.1.1) is not introducing
classes of new polytopes to Π(P ).
Remark 5.1.7. McMullen [56] originally defined Π(P ) as the subalgebra generated
by the classes of Minkowski summands of P . The following result of Shephard [47,
Section 15.2.7] implies that both definitions are equivalent: a polytope Q is a
deformation of P if and only if for small enough λ > 0, λQ is a Minkowski summand
of P .
The relations between [Q] and log[Q] in (5.1.3) and (5.1.4) show that Π(P ) is
generated by homogeneous elements. Thus, the grading of Π(V ) induces a grading
of Π(P ). We let Ξr (P ) = Π(P ) ∩ Ξr (V ) denote the component of Π(P ) in degree r.
Unlike the full algebra Π(V ), the subalgebra Π(P ) has finite dimension. McMullen
described the dimension of the graded components of Π(P ) when P is a simple
polytope.
Theorem 5.1.8 ([56, Theorem 6.1]). Let P be a d-dimensional simple polytope.
86
Then,
dimR (Ξr (P )) = hr (P )
for r = 0, 1, . . . , d.
Let F be a face of P and v ∈ relint(NF P ). The maximization operator x 7→ xv
defines a morphism of graded algebras
ψF : Π(P ) → Π(F )
(5.1.8)
that only depends on the face F and not on the particular choice of v ∈
relint(NF P ).
First observe that this map is well defined; that is, [Qv ] ∈ Π(F ) for every
generator [Q] of Π(P ). Indeed, if Q is a summand of P , say P = Q + Q0 , then
F = Pv = Qv + Q0v , so Qv is a Minkowski summand of F . Moreover, since the
normal fan of P refines that of Q, then Qw = Qv for any other w ∈ relint(NF P ).
Therefore the morphism (5.1.8) only depends on F .
Theorem 5.1.9 ([56, Theorem 2.4]). Let P be a simple polytope and F a face
of P . Then, the morphism ψF is surjective.
5.2
The polytope algebra as a module
Fix a hyperplane arrangement A in V . Take a normal vector vH for each hyperplane
H ∈ A, and consider the zonotope (Minkowski sum of segments):
Z=
X
Conv{0, vH }.
H
87
(5.2.1)
Its normal fan ΣZ coincides with the collection of faces Σ[A] of the arrangement
A. We say that a polytope Q is a generalized zonotope of A it is a deformation
of Z. In this section, we will work with the Tits algebra with real coefficients.
5.2.1
The module structure
We now consider the algebra Π(Z) introduced in Section 5.1.2. It is generated by
the classes of generalized zonotopes of A. It only depends on the arrangement A
and not on the particular choice of normal vectors vH . We start with a simple yet
interesting result. Recall the morphism ψF : Π(Z) → Π(F ) defined for every face
F ≤ Z in (5.1.8), it sends the class [Q] of a generalized zonotope of A to [Qv ]
where v ∈ V is any vector such that F = Zv .
Proposition 5.2.1. Let Z be a zonotope, and F a face of Z. Then, the morphism ψF is surjective.
Proof. Let F be a face of Z, F ∈ Σ[A] be its normal cone, and X = s(F ) be the
flat orthogonal to F . It follows from (5.2.1) that F = ZF is a translate of
ZX :=
X
Conv{0, vH }.
H : H⊇X
In particular, F is a Minkowski summand of Z. Being a Minkowski summand is
a transitive relation. Hence, any generator [Q] of Π(F ) is also in Π(Z). That
is, Π(F ) is a subalgebra of Π(Z). Moreover, if v ∈ relint(F ) and Q is a Minkowski
ψ
F
summand of F , then Qv = Q. Therefore, the composition Π(F ) ,→ Π(Z) −→
Π(F )
is the identity map. Consequently, the morphism ψF is surjective.
Remark 5.2.2. Compare with Theorem 5.1.9 and note that we do not assume the
zonotope Z to be simple. For an arbitrary polytope P and a face F of P , there
88
is no natural morphism Π(F ) → Π(P ), unlike in the previous case. This is a
particular property of zonotopes. Indeed, a polytope P is a zonotope if and only
if every face F ≤ P is a Minkowski summand of P , see [28, Proposition 2.2.14] for
a proof.
Let F be a face of A and F = ZF the corresponding face of Z. We define
right multiplication by the basis element HF ∈ RΣ[A] on Π(Z) as the projection
ψF : Π(Z) → Π(F ) ⊆ Π(Z).
Theorem 5.2.3. The algebra Π(Z) is a right RΣ[A]-module under the action
above. Explicitly, for a generator [Q] of Π(Z) and a basis element HF of RΣ[A],
[Q] · HF := [Qv ],
where v ∈ relint(F ).
Moreover, each graded component Ξr (Z) is a RΣ[A]-submodule and the action of
basis elements {HF }F on Π(Z) is by (graded) algebra endomorphisms.
Proof. The zero vector belongs to the central face O, so the action is clearly unital. Associativity follows from the following fact about polytopes [47, Section
3.1.5]. If Q ⊆ V is a polytope and v, w ∈ V , then (Qv )w = Qv+λw for any
small enough λ > 0. Similarly, the definition of the Tits product is such that
if v ∈ relint(F ) and w ∈ relint(G), then v + λw ∈ relint(F G) for any small
enough λ > 0. Hence,
([Q] · HF ) · HG = [(Qv )w ] = [Qv+λw ] = [Q] · HF G .
It follows that this product gives Π(Z) the structure of a right RΣ[A]-module.
The second statement follows directly from Theorem 5.1.5 and the characterization of the graded components Ξr in (5.1.6). Indeed, for any x ∈ Ξr (Z) and
89
λ > 0,
δλ (x · HF ) = δλ (x) · HF = λr x · HF = λr (x · HF ),
thus x · HF ∈ Ξr (Z).
5.2.2
Eulerian idempotents and diagonalization
An Eulerian family of A is a collection of idempotent and mutually orthogonal
elements {EX }X∈L[A] ⊆ RΣ[A] of the form
EX =
X
aF HF ,
(5.2.2)
F : s(F )≥X
with aF 6= 0 for at least one F with s(F ) = X. It follows that {EX }X is a complete
system of primitive orthogonal idempotents and that s(EX ) = QX [8, Theorem
11.20]. That is,
EX EY =



E
if X = Y,


0
otherwise,
X
X
EX = HO ,
X
and EX cannot be written as the sum of two non-trivial idempotents. The following
important property of Eulerian idempotents is [8, Lemma 11.12]. It was first proved
by Saliola [65, Lemma 1.4] in for particular families of Eulerian idempotents.
Lemma 5.2.4. For any face F and flat X, if s(F ) 6≤ X, then HF · EX = 0.
Example 5.2.5. Let C2 be the coordinate arrangement in R2 . The following is an
Eulerian family of C2 . Observe that in this example only faces in the first quadrant
have non-zero coefficients.
−1
1
E⊥ =
1
−1
−1
1
−1
Ex=0 =
Ey=0 =
90
1
1
E> =
A characteristic element w of non-critical 1 parameter t uniquely determines an
Eulerian family E = {EX }X , which satisfies
w=
X
tdim(X) EX .
(5.2.3)
X
This is a consequence of [8, Propositions 11.9, 12.59]. It follows that the action of
such a characteristic elements w on any RΣ[A]-module M is diagonalizable.
Let M be a (right) RΣ[A]-module, w ∈ RΣ[A] be a characteristic element of
non-critical parameter t and {EX }X be the corresponding Eulerian family. Then,
we have a decomposition
M=
M
M · EX
(5.2.4)
X
of vector spaces. Expression (5.2.3) shows that w acts on M · EX by multiplication
by tdim(X) . We define
ηX (M ) := dimR (M · EX ).
The kernel of the support map s is precisely the radical of RΣ[A] [8, Proposition
9.22]. Consequently, the character χM : RΣ[A] → R of M factors through RL[A]:
RΣ[A]
χM
s
RL[A]
χM
R
Thus, ηX (M ) = dimR (M · EX ) = χM (EX ) = χM (QX ) is independent of the characteristic element w. Furthermore, using relations (3.1.1) and the linearity of χM we
deduce
ηX (M ) = χM (QX ) =
X
µ(X, Y)χM (HY )
Y≥X
=
X
µ(X, Y)χM (HFY ) =
Y≥X
1
X
µ(X, Y) dimR (M · HFY ) (5.2.5)
Y≥X
t ∈ R is non-critical if it is not a root of χ(AX , t) for any flat X ∈ L[A].
91
where FY ∈ Σ[A] is such that s(FY ) = Y. The last equality follows since HFY is an
idempotent element, and thus χM (HFY ) = dimR (M · HFY ).
Moreover, the number of composition factors Mi+1 /Mi isomorphic to the simple
module indexed by X in a composition series 0 ⊂ M1 ⊂ M2 ⊂ · · · ⊂ Mk = M of M
is precisely ηX (M ). See [8, Section 9.5 and Theorem D.37] for details.
5.2.3
Simultaneous diagonalization
Let λ > 0 and let w ∈ RΣ[A] be a characteristic element of non-critical parameter
t. We know that the dilation morphism δλ and the action of w are diagonalizable. Moreover, since δλ and the action of w commute, they are simultaneously
diagonalizable. A natural question is to determine the eigenvalues of δλ and of the
action of w in their simultaneous eigenspaces. We completely answer this question
in the case of the Coxeter arrangements of type A and B in the next two sections.
The following result holds in the general case.
Proposition 5.2.6. Let x ∈ Π(Z) be a (nonzero) simultaneous eigenvector for δλ
and w with eigenvalues λr and tk , respectively. Then, r + k ≤ d.
Proof. Let {EX }X be the Eulerian family associated to w. Using the characterization of the graded components Ξr as the eigenspaces of δλ , and the decomposition
of RΣ[A]-modules in (5.2.4), we deduce that the common eigenspace for δλ and w
L
with the given eigenvalues is X Ξr (Z)·EX , where the sum is over all k-dimensional
flats of A. Without loss of generality, we assume that x ∈ Ξr (Z) · EX for a single
k-dimensional flat X.
Proposition 5.2.1 implies that Ξr (Z) · HF = Ξr (ZY ), where F ∈ Σ[A] is any face
92
of support Y. Hence, formula (5.2.5) yields
X
ηX (Ξr (Z)) =
µ(X, Y) dimR (Ξr (ZY )).
(5.2.6)
Y≥X
If k > d − r, then dim(ZY ) = d − dim(Y) ≤ d − k < r and dimR (Ξr (ZY )) = 0 for
any flat Y in the sum. So, in this case, dimR (Ξr (Z) · EX ) = ηX (Ξr (Z)) = 0. This
contradicts that x ∈ Ξr (Z) · EX is a nonzero element. Therefore, r + k ≤ d.
We have shown that
M
Ξr (Z) · EX
dim(X)=k
is the common eigenspace for δλ and w with eigenvalues λr and tk , respectively.
It turns out that, if the characteristic element w is projective (wF = wF ), elements in this subspace are also eigenvectors for negative dilations and for the Euler
map (5.1.7).
Proposition 5.2.7. Suppose w is projective. Then, for all x ∈
M
Ξr (Z) · EX ,
dim(X)=k
x∗ = (−1)d−k x
and
δ−1 (x) = (−1)d−k−r x.
Proof. In view of Theorem 5.1.6, the two claims are equivalent. We prove the first
one.
Since dilations and the Euler map are linear, and Ξr (Z) is generated by elements
of the form (log[P ])r , we can assume x = (log[P ])r · EX for a fixed generalized
zonotope P and a fixed k-dimensional flat X. Moreover, with EX as in (5.2.2), we
have
r
r
(log[P ]) · EX = (log[P ]) · EX · EX =
X
s(F )≥X
93
F
r
a (log[PF ])
· EX .
In the first equality we used that EX is idempotent. Consider a term aF (log[PF ])r
in the sum. Since (PF )F = PF , we have that NPF PF ≥ s(F ) ≥ X. If NPF PF > X,
take F 0 maximal inside NPF PF and note that
(log[PF ])r · EX = (log[PF ])r · HF 0 · EX = 0.
The last equality follows from Lemma 5.2.4. Thus, we have shown that (log[P ])r ·EX
can be written as a linear combination of elements of the form (log[Q])r · EX with
NQ Q = X. It is then enough to prove the claim for elements of this form, so we
now suppose NP P = X.
(log[P ])r · EX
∗
= (−1)r δ−1 (log[P ])r · EX
= (−1)r δ−1 (log[P ])r · EX
∗
= (log[P ])r · EX
X
=
(−1)dim(Q) (log[Q])r · EX
hTheorem 5.1.6i
hEX is projectivei
hTheorem 5.1.6i
Q≤P
= (−1)dim(P ) (log[P ])r · EX .
In the second equality we use that δ−1 (x · HF ) = δ−1 (x) · HF and that each EX
is projective if w is. The last equality follows again because for proper faces
Q < P , NQ Q > X and (log[Q])r · EX = 0. The result follows by observing that
dim(P ) = d − k.
If in addition A is a simplicial arrangement, like in the case of reflection arrangements, then Z and each of its faces are simple polytopes. In that case,
Theorem 5.1.8 allows us to replace dimR (Ξr (ZY )) by hr (ZY ) in expression (5.2.6).
Multiplying by z r and taking the sum over all values of r, we obtain
X
r
ηX (Ξr (Z))z r =
X
Y≥X
94
µ(X, Y)h(ZY , z).
(5.2.7)
5.2.4
First example: the cube and the coordinate arrangement
Let Cd be the coordinate arrangement in Rd . We identify the lattice of flats L[Cd ]
with the (opposite) boolean lattice 2[d] in the following manner:
S ⊆ [d] ←→ XS :=
\
{x : xi = 0}.
i∈S
Observe that XS ≤ XT if and only if T ⊆ S, and in this case µL (XS , XT ) =
µ2[d] (T, S) = (−1)|S\T | .
The d-cube d = [0, 1]d a zonotope of Cd . It is the Minkowski sum of the d
line segments li := Conv{0, ei } for i = 1, . . . , d. It is a simple polytope with hpolynomial h(d , z) = (1 + z)d . Furthermore, for any S ⊆ [d] we have
(d )XS =
X
li ∼
= |S| .
i∈S
Let us consider the right RΣ[Cd ]-module Π(d ). For a flat XS , formula (5.2.7)
yields
X
ηXS (Ξr (d ))z r =
r
X
µ(XS , XT )h(|T | , z) =
T ⊆S
X
(−1)|S\T | (1 + z)|T | = z |S| .
T ⊆S
Hence,
ηXS (Ξr (d )) =



1 if |S| = r,


0 otherwise.
In particular, a series decomposition of Ξr (d ) contains exactly one copy of the
simple module indexed by XS for every S ∈ [d]
.
r
Let us now consider the characteristic element γt ∈ RΣ[Cd ] in Section 3.5.4 for
95
t 6= 1. It is defined by
γt =
X
γtF HF ,
where γtF =
F



(t − 1)dim(F )
if F lies in the first orthant,


0
otherwise.
For each S ⊆ [d], let FS be the intersection of the first orthant with XS , it is a
face of Cd . We have s(FS ) = XS and T ⊆ S if and only if FS ≤ FT . A simple
computation shows that the Eulerian family corresponding to the characteristic
element γt is determined by
EXS =
X
(−1)|S\T | HFT .
T ⊆S
In dimension 2, this is the Eulerian family in Example 5.2.5. For each S ⊆ [d],
define
yS =
Y
log[li ] ∈ Π(d ).
i∈S
Example 5.1.2 shows that yS is a nonzero element of Π(d ).
We claim that {yS }S⊆[d] is a basis of simultaneous eigenvectors of Π(d ). Explicitly, yS is an eigenvector for the action of γt of eigenvalue td−|S| , and for
the action of δλ of eigenvalue λ|S| (λ > 0).
The second statement is clear,
since log[li ] ∈ Ξ1 (d ). Moreover, using that log[li ] = [li ] − 1, we have
yS =
Y
([li ] − 1) =
i∈S
X
(−1)|S\T | [XT ].
T ⊆S
On the other hand, observe that
[XS ] · EXS =
X
(−1)|S\T | [XS ] · HFT =
T ⊆S
X
(−1)|S\T | [XT ] = yS .
T ⊆S
Therefore,
yS ∈ Ξr (d ) ∩ (Π(d ) · EXS ) = Ξr (d ) · EXS .
The claim follows since dim(XS ) = d − |S|.
96
5.2.5
The zonotope module of a product of arrangements
The Cartesian product of two arrangements A in V and A0 in W is the following
collection of hyperplanes in V ⊕ W :
A × A0 = {H ⊕ W : H ∈ A} ∪ {V ⊕ H : H ∈ A0 }.
One can easily verify that Σ[A × A0 ] ∼
= Σ[A] × Σ[A0 ] as monoids. Hence, RΣ[A ×
A0 ] ∼
= RΣ[A] ⊗ RΣ[A0 ]. In fact, it is also true that
Π(Z × Z 0 ) ∼
= Π(Z) ⊗ Π(Z 0 ),
where Z and Z 0 are zonotopes of A and A0 , respectively, and therefore Z × Z 0 is a
zonotope of A × A0 . Indeed, every generalized zonotope of A × A0 is the Cartesian
product of generalized zonotopes of A and A0 . The corresponding isomorphism is
induced by
Π(Z) ⊗ Π(Z 0 ) → Π(Z × Z 0 )
[P ] ⊗ [Q]
7→ [P × Q]
The fact that this map is well-defined and a morphism of RΣ[A × A0 ]-modules
follows from the ideas in Section 5.5.1.
5.3
The module of generalized permutahedra
Generalized permutahedra are the deformations of the standard permutahedron
πd ⊆ Rd . Edmonds first introduced them under a different name in [38], where he
studied their relation to submodular functions and optimization. For a thorough
study of the combinatorics of these polytopes, see [1, 60, 61].
In this section, we study the algebra Π(πd ) of generalized permutahedra and
its structure as a module over the Tits algebra of the braid arrangement Ad . We
97
begin with a brief review of the braid arrangement, its relation with the symmetric
group, and some statistics on permutations.
5.3.1
The symmetric group and the Eulerian polynomial
The symmetric group Sd is the group of permutations σ : [d] → [d] under composition. It is the Coxeter group corresponding to the braid arrangement. It acts
on Rd by permuting coordinates:
σ(x1 , x2 , . . . , xd ) = (xσ(1) , xσ(2) , . . . , xσ(d) ).
For a permutation σ ∈ Sd , we let s(σ) denote the subspace of points fixed by
the action of σ; it is a flat of Ad . In view of the identification between flats of Ad
and partitions of [d], s(σ) can equivalently be defined as the partition of [d] into
the disjoint cycles of σ. For example, if in cycle notation σ = (13)(2658)(4)(7),
then s(σ) = {13, 2568, 4, 7}.
Recall that i ∈ [d−1] is a descent of σ ∈ Sd if σ(i) > σ(i+1), and i ∈ [d−1] is
an excedance of σ ∈ Sd if σ(i) > i. Let des(σ) and exc(σ) denote the number of
descents and excedances of σ, respectively. In the example above, 1, 2, 5 (in bold)
are the excedances of σ, and we have exc(σ) = 3. We can similarly define descents
and excedances for permutations of any set S with a total order ≺, we denote the
corresponding statistics by des≺ and exc≺ .
It is a classical result that descents and excedances are equidistributed in Sd .
That is,
Ad,k := {σ ∈ Sd : des(σ) = k} = {σ ∈ Sd : exc(σ) = k} ,
for all possible values of k. Foata’s fundamental transformation provides a simple
98
proof of this result. The numbers Ad,k are the classical Eulerian numbers (OEIS:
A008292). The Eulerian polynomial Ad (z) is:
Ad (z) :=
d−1
X
Ad,k z k =
X
z exc(σ) .
σ∈Sd
k=0
The exponential generating function for these polynomials was originally given by
Euler himself:
A(z, x) = 1 +
X
Ad (z)
d≥1
z−1
xd
=
.
d!
z − ex(z−1)
(5.3.1)
See [40, Section 3] for a derivation of this formula.
Let C(S) the collection of cyclic permutations on a finite set S, and C(d) =
C([d]). Given a permutation σ ∈ Sd and a block S ∈ s(σ), the restriction σ|S of σ to
S is a cyclic permutation. For example, with σ as before and S = {2, 5, 6} ∈ s(σ),
we have σ|S = (265) ∈ C({2, 5, 6}). A very simple but important observation is
that the number of excedances of σ can be computed by adding up the excedances
P
in each cycle in its cycle decomposition. That is, exc(σ) = S∈s(σ) exc(σ|S ). The
number of excedances in each cycle σ|S is computed with respect to the natural
order in S ⊆ [d].
5.3.2
The module Generalized permutahedra
The permutahedron πd ⊆ Rd is the convex hull of the Sd -orbit the point
(1, 2, . . . , d). It is a zonotope of the braid arrangement Ad and has dimension d−1.
Deformations of πd are called generalized permutahedra. We consider the module Π(πd ) as in Section 5.2. The main goal of this section will be to prove the
following result.
99
Theorem 5.3.1. For any flat X ∈ L[Ad ] and r = 0, 1, . . . , d − 1,
ηX (Ξr (πd )) = {σ ∈ Sd : s(σ) = X, exc(σ) = r} .
(2, 3, 1)
(1, 3, 2)
(3, 2, 1)
(1, 2, 3)
(3, 1, 2)
(2, 1, 3)
Figure 5.3.1: The permutahedron in R3 and R4 .
The relation between Π(πd ) and statistics on Sd is via the h-polynomial of
πd . Brenti [30, Theorem 2.3] showed that h(πd , z) = Ad (z). Moreover, for a
flat/partition X = {S1 , . . . , Sk } of Ad , the face (πd )X is a translate of π|S1 | × · · · ×
π|Sk | , a product of lower-dimensional permutahedra. Thus,
h((πd )X , z) = A|S1 | (z) · . . . · A|Sk | (z).
(5.3.2)
The next lemma is an essential ingredient in the proof of Theorem 5.3.1.
Lemma 5.3.2. For every d ≥ 1,
X
µ(⊥, X)A|S1 | (z) · . . . · A|Sk | (z) =
X={S1 ,...,Sk }`[d]
X
z exc(σ) .
(5.3.3)
σ∈C(d)
Proof. We will show that the exponential generating function of both sides
of (5.3.3) are equal to log(A(z, x)), where A(z, x) is the generating function for
the Eulerian polynomials in (5.3.1).
First, recall that if X = {S1 , . . . , Sk }, then µ(⊥, X) = (−1)k−1 (k − 1)!. Thus,
a direct application of the Compositional Formula [68, Theorem 5.1.4] shows that
100
the exponential generating function of the LHS of (5.3.3) is the composition of
X
(−1)d−1 (d − 1)!
d≥1
xd
= log(1 + x)
d!
X
with
Ad (z)
d≥1
xd
= A(z, x) − 1,
d!
which is precisely log(A(z, x)).
On the other hand, grouping permutations with the same underlying partition s(σ), we obtain
A(z, x) = 1 +
X X
d≥1
z
exc(σ)
xd
d!
σ∈Sd
=1+
X X X
d≥1
z
exc(σ)
xd
σ∈Sd
s(σ)=X
X`[d]
d!
.
Since a permutation σ with s(σ) = X is the product of cyclic permutations σS ∈
P
C(S) for each block S ∈ X, and in this case exc(σ) = S∈X exc(σS ),
X
z
exc(σ)
=
Y X
S∈X
σ∈Sd
s(σ)=X
z
exc(σS )
.
(5.3.4)
σS ∈C(S)
Thus, the Exponential Formula [68, Corollary 5.1.6] implies that
A(z, x) = exp
X X
d≥1
z
exc(σ)
σ∈C(d)
xd d!
.
Taking logarithms on both sides yields the result.
A small modification in the proof of the previous Lemma immediately gives the
following result, which was first discovered by Brenti.
Corollary 5.3.3 ([31, Proposition 7.3]). The following identity holds
1+
X X
d≥1
σ∈Sd
t
| s(σ)| exc(σ)
z
xn
n!
= exp(t log(A(z, x))) =
z−1
z − ex(z−1)
t
.
An analogous formula for the type B Coxeter group is described in Proposition 5.4.7. We are now ready to prove the main result of this section.
101
Proof of Theorem 5.3.1. We will compute the values ηX (Ξr (πd )) using formula
(5.2.7), which in this case reads
X
ηX (Ξr (πd ))z r =
r
X
µ(X, Y)h((πd )Y , z).
Y : Y≥X
Using (2.2.6) and (5.3.2), we can rewrite the expression above as
X
r
ηX (Ξr (πd ))z =
r
Y
S∈X
X
µ(⊥, Y)A|T1 | (z) · . . . · A|T` | (z) .
Y={T1 ,...,T` }`S
Now, an application of Lemma 5.3.2 and relation (5.3.4) gives
X
r
ηX (Ξr (πd ))z =
r
Y X
S∈X
σ∈C(S)
z
exc(σ)
=
X
z exc(σ) .
σ∈Sd
s(σ)=X
Finally, taking the coefficient of z r on both sides of the last equality yields the
result.
Adding over all flats with the same dimension in Theorem 5.3.1, we conclude
the following.
Corollary 5.3.4. Let w ∈ RΣ[Ad ] be a characteristic element of non-critical parameter t and λ > 0. The dimension of the simultaneous eigenspace for w and δλ
with eigenvalues tk and λr is
{σ ∈ Sd : | s(σ)| = k, exc(σ) = r} .
5.3.3
Simultaneous-eigenbasis for the Adams element
Let αt ∈ RΣ[Ad ] be the Adams element of parameter t as in Section 3.5.1. It is
invariant with respect to the action of Sd , and its action on RΣ[Ad ]-modules is
closely related with the convolution powers of the identity map of a Hopf monoid,
102
see [7, Section 14.4] and 6.3.3. The corresponding Eulerian idempotents are [8,
Theorem 12.75]
EX =
X X (−1)dim(G/F )
1
HG ,
dim(X)!
deg(G/F
)
G≥F
(5.3.5)
s(F )=X
where dim(G/F ) = dim(G) − dim(F ) and, deg(G/F ) =
t
2
Q
S∈F
G|S .
t
2
t
3
αt =
− 12
E⊥ =
1
3
− 12
− 12
− 12
t
3
t
3
t
2
= tE⊥ + t2 E12,3 + t2 E13,2 + t2 E23,1 + t3 E>
t
2
− 21
1
2
1
3
1
3
1
3
t
2
t
3
t
1
1
3
t
3
3
t
2
1
3
t
1
6
− 14
1
E12,3 = − 4
− 21
− 14
− 41
1
2
E> =
1
6
1
6
1
6
1
6
1
6
Figure 5.3.2: The Adams element αt and some of the associated Eulerian idempotents of the braid arrangement in R3 .
Theorem 5.3.1 suggest the existence of a natural basis for Ξr (πd )·EX indexed by
permutations σ with r excedances and s(σ) = X. In this section we will construct
a candidate for such basis.
The standard simplex ∆[d] ⊆ Rd is the convex hull of the standard basis {e1 , . . . , ed } of Rd . Similarly, for any nonempty subset S ⊆ [d], we let ∆S =
Conv{ei : i ∈ S}. Ardila, Benedetti and Doker showed [13, Proposition 2.4] that
every generalized permutahedron P can be written uniquely as a signed Minkowski
P
yS ∆S . This means that we have the following identity:
sum of simplices P =
P+
X
|yS |∆S =
yS <0
X
yS >0
103
yS ∆S .
Up to translation, this is equivalent to the following identity in Ξ1 (πd ):
log[P ] =
X
yS log[∆S ].
S⊆[d]
Given that log[∆S ] = 0 whenever S is a singleton, and that [P ] = [Q] if and
only if P is a translate of Q, we conclude that {log[∆S ] : S ⊆ [d], S ≥ 2} is
a linear basis for Ξ1 (πd ) and therefore generate Π(πd ) as an algebra. This agrees
with dimR (Ξ1 (πd )) = h1 (πd ) = 2d − d − 1.
We will use a bijection between increasing rooted forests on [d] and permutations in Sd . An increasing rooted forest is a disjoint union of planar rooted trees
where each child is larger than its parent and the children are in increasing order
from left to right. Given a rooted forest t, the corresponding permutation σ(t) is
read as follows. Each connected component of t corresponds to a cycle of σ(t). To
form a cycle, traverse the corresponding tree counterclockwise and record a node
the second time you pass by it2 .
1
2
4 10
3 5 9 11 12
6 8
7
7−→
(3 7 6 8 5 2 9 4 11 12 10 1)
The inverse can be described inductively by writing each cycle with its minimum
element in the last position, and using right to left minima. We omit the details,
but provide an example σ 7→ t(σ) to illustrate the idea.
1
(7 3 6 9 5 1)(4 10 8 2) 7−→
1
2
(73) (695) 4 (10 8)
3
7−→
7
2
5
6
4
9
8
10
(5.3.6)
2
A similar bijection is described by Peter Luschny in this OEIS entry.
104
This bijection is such that the connected components of the forest t(σ) are the
blocks of s(σ). Moreover, the number of leaves of t(σ) in S ∈ s(σ) is exc(σ|S ) (a
tree consisting only of its root has zero leaves). Consequently, the total number of
leaves of t(σ) is exc(σ).
Let σ ∈ Sd be a permutation with r excedances and let X = s(σ). For 1 ≤ i ≤ r,
let Ji be the elements on the path from the ith leaf of t(σ) to the root of the
corresponding tree. Define the element
r
Y
xσ =
log[∆Ji ] · EX .
(5.3.7)
i=1
For instance, if σ is the permutation in (5.3.6), then
xσ = log[∆{7,3,1} ] log[∆{6,5,1} ] log[∆{9,5,1} ] log[∆{4,2} ] log[∆{10,8,2} ] · EX ,
where X = {1, 3, 5, 6, 7, 9}, {2, 4, 8, 10}.
Conjecture 5.3.5. For fixed X ` [d] and r ≤ d − X , the collection
{xσ : s(σ) = X, exc(σ) = r}
is a linear basis of Ξr (πd ) · EX .
It follows from the definition (5.3.7) that xσ ∈ Ξr (πd ) · EX . The content of the
conjecture is that these elements are linearly independent. Explicit computations
show that this is the case for d = 2, 3, 4. Proposition 5.3.6 Proposition 5.3.7 below
prove the extremal cases r = 1 and r = d − |X| of this conjecture, respectively.
Proposition 5.3.6. For a subset J ⊆ [d] of cardinality at least 2, let XJ ` [d] be
the partition whose only non-singleton block is J. Then,
log[∆J ] · EXJ
is a nonzero element. Furthermore, {log[∆J ] · EXJ : J ⊆ [d], |J| ≥ 2} is a basis of
simultaneous eigenvectors for Ξ1 (πd ).
105
Proof. First, observe that any cyclic permutation on a set with more than one
element has at least one excedance, and only one cyclic permutation attains this
minimum. Namely, the only cyclic permutation in Sd having one excedance is
(d d − 1 . . . 2 1). Hence, a permutation σ ∈ Sd has at least as many excedances as
non-singleton blocks in s(σ). It then follows from Theorem 5.3.1 that



1 if X ` [d] has exactly one non-singleton block,
dimR (Ξ1 (πd ) · EX ) =


0 otherwise.
Thus, the second statement follows from the first.
Since {log[∆J ] : J ⊆ [d], J ≥ 2} is a linear basis for Ξ1 (πd ), it is enough
to write log[∆J ] · EXJ as a non-trivial linear combination of these basis elements.
Observe that if F = (S1 , S2 , . . . , Sk ), then [∆J ] · HF = [∆J∩Si ] where i is the first
index for which the intersection J ∩ Si is nonempty. Thus,



[∆J ]
if s(F ) ≤ XJ ,
[∆J ] · HF =


[a proper face of ∆J ] otherwise.
Using that the action of HF is an algebra morphism, we have log[∆J ]·HF = log([∆J ]·
HF ). Hence, the coefficient of log[∆J ] in log[∆J ] · EXJ is
X
1
1 = 1.
dim(XJ )!
s(F )=XJ
The equality follows since for any flat X of the braid arrangement, AX
d has dim(X)!
chambers.
Note that the element log[∆J ]·EXJ in the proposition is precisely the element xσ
for the unique permutation σ with s(σ) = XJ and exc(σ) = 1. Indeed, t(σ) consists
of a increasing path whose nodes are the elements in J and isolated roots indexed
by the elements in [d] \ J.
106
Proposition 5.3.7. For any X = {S1 , . . . , Sk } ` [d], the space Ξd−k (πd ) · EX is 1dimensional. Moreover,
xX =
k Y
i=1
Y
log[∆{min(Si ),j} ]
(5.3.8)
j6=min(Si )
is a nonzero element in Ξd−k (πd ) · EX .
Proof. Observe that any cyclic permutation on a set with s elements has at most s−
1 excedances, and only one cyclic permutation attains this maximum. Namely, the
only cyclic permutation in Sd having d − 1 excedances is (1 2 . . . d − 1 d). Hence,
for any X = {S1 , . . . , Sk } ` [d] there is exactly one permutation with s(σ) = X
and d − k excedances. Theorem 5.3.1 then implies that dimR (Ξd−k (πd ) · EX ) = 1.
It follows from Example 5.1.2 that the element xX is nonzero, and counting the
number of factors in (5.3.8) shows that xX ∈ Ξd−k (πd ). Thus, we are only left to
prove that xX ∈ Ξd−k (πd ) · EX . That is, that xX · EX = xX
Let G ∈ Σ[Ad ] with s(G) > X. Then, for some block Si ∈ X and some a ∈ Si , a
and min(Si ) are not in the same block of s(G). Hence [∆{min(Si ),a} ] · HG is the class
of a point, and log[∆{min(Si ),a} ] · HG = log[{0}] = 0. Since the action of HG is an
algebra morphism, we get that xX · HG = 0. Therefore,
xX · EX = xX ·
X
X
1
1
HF =
xX · HF = xX .
dim(X)!
dim(X)!
s(F )=X
s(F )=X
The element xX in the previous result is xσ for the only permutation σ with
s(σ) = X and exc(σ) = d − |X|. In this case, the forest t(σ) consists of a k trees
with node sets S1 , . . . , Sk , respectively. Each tree has min(Si ) as a root and every
other element in Si as a child of the root.
107
5.4
The module of type B generalized permutahedra
In this section, we study the algebra Π(πdB ) of type B generalized permutahedra
and its structure as a module over the Tits algebra of the Coxeter arrangement of
type B.
5.4.1
The hyperoctahedral group and the Type B Eulerian
polynomial
The hyperoctahedral group Bd is the group of bijections σ : [±d] → [±d] satisfying σ(i) = σ(i) for all i ∈ [±d] under composition. Elements in Bd are called
signed permutations. The group Bd acts on Rd by permutation and sign changes
of coordinates:
σ(x1 , x2 , . . . , xd ) = (xσ(1) , xσ(2) , . . . , xσ(d) ).
Recall that, for instance, x1 = −x1 . For a signed permutation σ ∈ Bd , we let s(σ)
denote the subspace of points fixed by the action of σ; it is a flat of A±
d . Under the
identification above, s(σ) is the signed partition of [±d] obtained from the underlying the cycle decomposition of σ by merging all the blocks that contain an element i
and its negative i. For example, if in cycle notation σ = (1)(1̄)(22̄)(343̄4̄)(56̄)(5̄6),
then s(σ) = {22̄33̄44̄, 1, 1̄, 56̄, 5̄6}.
Let σ ∈ Bn . The restriction σ|S0 to the zero block S0 ∈ s(σ) is a signed permutation of S0 . Its action on R|S0 |/2 does not fix any nonzero vector, so s(σ|S0 ) =⊥.
For a nonzero block S ∈ s(σ), σ|S ∈ C(S) is a cyclic permutation of the elements
in S. The restriction σ|±S is again a signed permutation, and it is completely
determined by either σ|S or σ|S .
108
We present some statistics on signed permutations. For σ ∈ Bd , let
Des(σ) = {i ∈ [d − 1] ∪ {0} : σ(i) > σ(i + 1)}
des(σ) = Des(σ)
Exc(σ) = {i ∈ [d − 1] : σ(i) > i}
exc(σ) = Exc(σ)
Neg(σ) = {i ∈ [d] : σ(i) < 0}
neg(σ) = Neg(σ)
fexc(σ) = 2 exc(σ) + neg(σ),
where we set σ(0) = 0. Elements in the sets above are descents, excedances and
negations of σ, respectively. The last statistic is called the flag-excedance of a
signed permutation. We define one last statistic, the B-excedance of σ:
c = exc(σ) + b neg(σ)+1
c.
excB (σ) = b fexc(σ)+1
2
2
(5.4.1)
Foata and Han [41, Section 9] show that descents and B-excedances are equidistributed. That is,
Bd,k := {σ ∈ Bd : des(σ) = k} = {σ ∈ Bd : excB (σ) = k} ,
for all possible values of k. The numbers Bd,k are the Eulerian numbers of type
B (OEIS: A060187). The type B Eulerian polynomial Bd (z) is:
Bd (z) =
d
X
Bd,k z k =
X
z excB (σ) .
σ∈Bd
k=0
The exponential generating function of these polynomials is first due to Brenti [30,
Theorem 3.4]. We will be interested in the type B exponential generating
function of these polynomials:
B(z, x) = 1 +
X
d≥1
Bd (z)
xd
(1 − z)ex(1−z)/2
=
,
(2d)!!
1 − zex(1−z)
(5.4.2)
where (2d)!! is the double factorial (2d)!! = (2d)(2d−2) . . . 2 = 2d d!. Substituting x
by 2x one recovers Brenti’s original formula.
109
5.4.2
The module of type B Generalized permutahedra
The type B permutahedron πdB ⊆ Rd is the convex hull of the Bd -orbit of the
point (1, 2, . . . , d). It is full-dimensional and a zonotope of A±
d . We now consider
the module Π(πdB ). The main result of this section is the following.
Theorem 5.4.1. For any flat X ∈ L[A±
d ] and r = 0, 1, . . . , d,
ηX (Ξr (πdB )) = {σ ∈ Bd : s(σ) = X, excB (σ) = r} .
(−1, 2)
(1, 2)
(−2, 1)
(2, 1)
(−2, −1)
(2, −1)
(−1, −2)
(1, −2)
Figure 5.4.1: Type B permutahedron in R2 and R3 .
As in type A, the relation between Π(πdB ) and statistics on Bd is due to Brenti’s
result showing that h(πdB , z) = Bd (z). For a flat X = {S0 , S1 , S1 , . . . , Sk , Sk } of A±
d,
B
× π|S1 | × · · · × π|Sk | , a product of lowerthe face (πdB )X is a translate of π|S
0 |/2
dimensional permutahedra of type A and B, where exactly one factor is of type B.
Thus,
h((πdB )X , z) = B|S0 |/2 (z) · A|S1 | (z) · . . . · A|Sk | (z).
(5.4.3)
The following result is the analogous in type B of Lemma 5.3.2.
Lemma 5.4.2. For every d ≥ 1,
X
µ(⊥, X)B|S0 |/2 (z)·A|S1 | (z)·. . .·A|Sk | (z) =
X
σ∈Bd
s(σ)=⊥
X={S0 ,...,Sk ,Sk }`B [±d]
110
z excB (σ) . (5.4.4)
In the same spirit as the proof of Lemma 5.3.2, we will establish (5.4.4) by
comparing the type B exponential generating function of both sides of the equality.
An important tool in this proof is the following analog of the compositional formula
for type B generating functions. For a proof, see Proposition 6.6.4.
Proposition 5.4.3 (Type B Compositional Formula - power series). Let
f (x) = 1 +
X
fd
d≥1
xd
(2d)!!
g(x) = 1 +
X
gd
d≥1
xd
(2d)!!
a(x) =
X
d≥1
ad
xd
.
d!
If
h(x) = 1 +
X
hd
d≥1
xd
(2d)!!
where
X
hd =
f|S0 |/2 gk a|S1 | . . . a|Sk | ,
{S0 ,S1 ,S1 ,...,Sk ,Sk
}`B [±d]
then
h(x) = f (x)g(a(x)).
Taking gd = 1 in the Type B Compositional Formula we deduce the following.
Corollary 5.4.4 (Type B Exponential Formula). Let f (x) and a(x) be as before.
If
h(x) = 1 +
X
hd
d≥1
xd
(2d)!!
where
hd =
X
f|S0 |/2 a|S1 | . . . a|Sk | ,
X`B [±d]
then
h(x) = f (x) exp(a(x)/2).
In the proof of Theorem 5.3.1, we used that for (type A) permutations σ ∈
Sd , exc(σ) equals the sum of exc(σ|S ) as S runs through the blocks of s(σ). For
signed permutations, one easily checks this also holds for the statistics exc and neg.
However, it is not obvious at all that the same is true for excB , since its definition
uses the floor function.
111
Consider the order ≺ of the elements of any involution-exclusive subset S ⊆
[±d] defined by:
i ≺ j ⇐⇒




0 < i < j, or




(5.4.5)
i < 0 < j, or






j < i < 0.
Proposition 5.4.5. Let σ ∈ Bd and s(σ) = {S0 , S1 , S1 , . . . , Sk , Sk }. Then,
excB (σ) = excB (σ|S0 ) + exc≺ (σ|S1 ) + · · · + exc≺ (σ|Sk ),
where exc≺ (σ|Si ) is the number of usual (type A) excedances of σ|Si with respect to
the order ≺.
Proof. For i ≥ 1, write σ|Si = (j1 j2 . . . j` ) in cycle notation.
Since σ|Si =
(j1 j2 . . . j` ), negations of σ|±Si are in correspondence with changes of sign in the
sequence
j1 7→ j2 7→ · · · 7→ j` 7→ j1 .
It follows that
neg(σ|±Si ) = 2 · {j ∈ Si : j < 0 < σ(j)}
is an even number.
Observe that according to the three cases of definition (5.4.5), a ≺-excedance
of σ|Si corresponds to either




an excedance of σ|±Si occurring in Si , or




a negation of σ|±Si occurring in Si , or






an excedance of σ|±Si occurring in Si ,
112
respectively. Since exactly half of the negations of σ|±Si occur in Si , we deduce
that
exc≺ (σ|Si ) = exc(σ|±Si ) +
Thus, in view of (5.4.1) and using that
neg(σ|±Si )
2
neg(σ|±Si )
.
2
is always an integer,
excB (σ) = exc(σ) + b neg(σ)+1
c
2
P
X
neg(σ|S0 )+ i neg(σ|±Si )+1
c
= exc(σ|S0 ) +
exc(σ|±Si ) + b
2
i
= exc(σ|S0 ) +
X
exc(σ|±Si ) + b
neg(σ|S0 )+1
c
2
+
X neg(σ|±S
i
)
2
i
i
= excB (σ|S0 ) + exc≺ (σ|S1 ) + · · · + exc≺ (σ|Sk ).
Proof of Lemma 5.4.2. Recall that µ(⊥, X) = (−1)k (2k − 1)!!, where |X| = 2k + 1.
Observe that
X −1/2
xd
=
1+
(−1) (2d − 1)!!
xd = (1 + x)−1/2 .
(2d)!!
d
d≥0
d≥1
X
d
Using the Type B Compositional formula, we conclude that the type B exponential
generating function of the LHS of (5.4.4) is
B(z, x)
− 1,
B(z, x)(1 + (A(z, x) − 1))−1/2 − 1 = p
A(z, x)
where A(z, x) and B(z, x) are the generating functions in (5.3.1) and (5.4.2),
respectively. On the other hand, Proposition 5.4.5 shows that for each partition X = {S0 , S1 , S1 , . . . , Sk , Sk } `B [±d],
X
σ∈Bd
s(σ)=X
z excB (σ) =
X
z excB (σ0 )
k X
Y
i=1
σ0 ∈B(S0 )
s(σ0 )=⊥
z exc(σ) .
(5.4.6)
σ∈C(|Si |)
In the proof of Lemma 5.3.2, we showed that the (usual) exponential generating
function of the terms in the product is log(A(z, x)). An application of the type B
113
Exponential Formula and (5.4.6) yields
log(A(z, x)) xd X X
excB (σ)
exp
B(z, x) = 1 +
z
(2d)!!
2
σ∈B
d≥1
d
s(σ)=⊥
Dividing both sides by exp
log(A(z,x)) 2
p
=
A(z, x) and subtracting 1 yields the
result.
We are now ready to complete the proof of the main theorem in this section.
The steps of the proof mirror those of the type A result.
Proof of Theorem 5.4.1. We will again use formula (5.2.7) to compute the values
ηX (Ξr (πdB )). Recall that the formula reads
X
ηX (Ξr (πdB ))z r
r
X
µ(X, Y)h((πdB )Y , z).
Y≥X
Using formulas (2.2.7) and (5.4.3), we see that for a flat X = {S0 , . . . , Sk , Sk } this
expression equals
X
µ(⊥, Y)B|T0 |/2 A|T1 | . . . A|T` |
Y`B S0
Y={T0 ,...,T` ,T` }
·
k Y
i=1
µ(⊥, Yi )A|T1i | (z) · . . . · A|T`i | (z) .
X
Yi `Si
Yi ={T1i ,...,T`i }
Using Lemmas 5.3.2 and 5.4.2 in each factor, we deduce
X
ηX (Ξr (πdB ))z r
r
=
X
σ∈B(S0 )
s(σ)=⊥
z
excB (σ)
Y
k X
i=1
σ∈C(|Si |)
z
exc(σ)
=
X
z excB (σ) ,
σ∈Bd
s(σ)=X
where the last equality is (5.4.6). Finally, taking the coefficient of z r on both sides
yields the result.
114
Adding over all flats with the same dimension in Theorem 5.4.1, we conclude
the following.
Corollary 5.4.6. Let w ∈ RΣ[A±
d ] be a characteristic element of non-critical
parameter t and λ > 0. The dimension of the simultaneous eigenspace for w and
δλ with eigenvalues tk and λr is
{σ ∈ Bd : dim(s(σ)) = k, excB (σ) = r} .
As in the type A case, we can modify the proof of the previous Lemma to obtain
P
the generating function for the bivariate polynomials σ∈Bd tdim(s(σ)) z excB (σ) . To
the best of our knowledge, this is a new result.
Proposition 5.4.7. The following identity holds
1+
X X
dim(s(σ)) excB (σ)
t
z
σ∈Bd
d≥1
t−1
xn
2
(1 − z)ex(1−z)/2
z−1
.
=
(2n)!!
1 − zex(1−z)
z − ex(z−1)
Proof. We can slightly modify (5.4.6) to obtain
X
dim(X) excB (σ)
t
z
=
X
σ∈Bd
s(σ)=X
z
excB (σ0 )
σ∈B(S0 )
s(σ)=⊥
k Y
i=1
t
X
z exc(σ) .
σ∈C(|Si |)
Using the (type B) generating functions of the factors deduced in the proofs of
Lemmas 5.3.2 and 5.4.2, and the type B compositional formula, we deduce
1+
X X
d≥1
tdim(s(σ)) z excB (σ)
σ∈Bd
which equals B(z, x)A(z, x)
t−1
2 .
xn
t log(A(z, x)) B(z, x)
=p
exp
,
(2n)!!
2
A(z, x)
Substituting the expressions for A(z, x) and
B(z, x) in (5.3.1) and (5.4.2) yields the result.
Specializing t := 0 and x := 2x gives an alternative expression for the exponential generating function of the OEIS sequence A156919. In our context, these
115
coefficients count the number of signed permutations whose action on Rd has no
nonzero fixed point weighted by the statistic excB .
5.4.3
Two bases for type B generalized permutahedra
Faces of the standard simplex ∆[d] correspond to linearly independent rays of the
cone of generalized permutahedra in Rd . In the language of the present work, this
means that {log[∆S ] : S ⊆ [d], |S| ≥ 2} forms a linear basis of Ξ1 (πd ) and that
log[∆S ] is not the sum of the log-classes of other generalized permutahedra (other
than trivial dilations of itself). The goal of this section is to establish an analogous
result for the type B case.
The standard simplex coincides with the weight polytope PSd (λ1 ) in the sense
of Ardila, Castillo, Eur, and Postnikov [14], where λ1 is the fundamental weight
(1, 0, . . . , 0) of Sd . In this manner, the cross-polytope PBd (λ1 ) = Conv{±ei : i ∈
[d]} is the type B analog of the standard simplex. However, in the same paper the
authors point out that the faces of the cross-polytope span a space of roughly half
the desired dimension. This is intuitively clear once we notice that the collection of
faces of PBd (λ1 ) entirely contained in one orthant span the same space in Ξ1 (πdB )
as the faces contained in the opposite orthant. The following result shows that
it is not possible to find a single type B generalized permutahedron whose faces
generate Ξ1 (πdB ).
Proposition 5.4.8. Let P = {Pα }α be a collection of type B generalized permutahedra such that {log[Pα ]}α spans Ξ1 (πdB ). Then, P contains at least 2d−1 full
dimensional polytopes.
±
Proof. Let {EX }X ⊆ RΣ[A±
d ] be an Eulerian family of Ad . The result follows from
116
the following two facts, which we justify below.
1. The projection of log[P ] to the subspace Ξ1 (πdB ) · E⊥ is zero unless P is
full-dimensional.
2. dimR (Ξ1 (πdB ) · E⊥ ) = 2d−1 .
1. Let P be a type B generalized permutahedron that is not full-dimensional.
Then we can choose a face F 6= O of A±
d such that PF = P , for instance any
maximal face in NP P . It follows from Lemma 5.2.4 that HF · E⊥ = 0. Therefore,
log[P ] · E⊥ = log([P ] · HF ) · E⊥ = log[P ] · (HF · E⊥ ) = 0.
2. Recall that by definition, dimR (Ξ1 (πdB ) · E⊥ ) = η⊥ (Ξ1 (πdB )). Using Theorem 5.4.1, this is the number of signed permutations σ ∈ Bd with s(σ) =⊥ and
excB (σ) = 1. A signed permutation σ ∈ Bd with s(σ) =⊥ is the product of cycles
on involution-inclusive subsets S ⊆ [±d]. Moreover, each such cycle adds at least
1 to the number of negations of σ. Thus, a signed permutation with s(σ) =⊥
and excB (σ) = 1 must be the product of either 1 or 2 cycles and have no excedances. Such cycles are necessarily of the form (d d − 1 . . . 1d d − 1 . . . 1). Thus,
the permutations counted by η⊥ (Ξ1 (πdB )) are in correspondence with unordered
pairs {S, [±d] \ S} of involution-inclusive subsets of [±d], and there are precisely
2d−1 many of them.
Alternatively, one can manipulate the generating function in Proposition 5.4.7
(differentiate with respect to z and specialize t = z = 0) to deduce that the
coefficient of t0 z 1 xd is 2d−1 .
In [59], Padrol, Pilaud, and Ritter construct a family of type B generalized
117
permutahedra called shard polytopes. They show that any type B generalized permutahedron can be written uniquely as a signed Minkowski sum of these polytopes
(up to translation). Already in R3 , there are 14 full-dimensional shard polytopes.
We proceed to construct a family of generators that achieves the minimum imposed
by the previous proposition.
For a nonempty involution-exclusive subset S ⊆ [±d], define the simplices
∆0S = Conv({0} ∪ {ei : i ∈ S}),
∆S = Conv{ei | i ∈ S}
where for i ∈ [d], ei = −ei . Observe that the only full-dimensional simplices in
this collection are ∆0S with |S| = d. We say that an involution-exclusive subset
S ⊆ [±d] is special if in addition min{|i| : i ∈ S} ∈ S. Observe that for any
nonempty involution-exclusive subset S ⊆ [±d], exactly one of {S, S} is special.
Theorem 5.4.9. Every type B generalized permutahedron can be written uniquely
as a signed Minkowski sum of the simplices ∆S and ∆0S with S special.
e3
e2
e1
Figure 5.4.2: The type B generators of Theorem 5.4.9 in R2 . In R3 , we only show
the 4 = 23−1 full-dimensional generators.
Remark 5.4.10. Observe that the generating collection {∆S : S ⊆ [d]} for generalized permutahedra is invariant under the action of Sd . In contrast, the collection
of generators for type B generalized permutahedra presented in the previous theorem fails to be invariant under the action of Bd . This is not by accident. Already
in R2 , we see that any collection of type B generalized permutahedra that contains
118
a triangle P (full-dimensional simplex) and that is invariant under the action of
Bd , will contain the rotations of P by 90◦ , 180◦ , and 270◦ , all of which are necessarily different. Thus, such a collection will not attain the minimum number of
full-dimensional polytopes required by Proposition 5.4.8.
Before proving the previous theorem, we exhibit a different generating collection
of type B generalized permutahedra that is invariant under the action of Bd .
Then, we will show that the collection if the previous theorem generates the same
collection of polytopes.
Given a collection of real numbers {zS }S ⊆ R, one for each nonempty
involution-exclusive subset S ⊆ [±d], consider the polytope
n
o
X
P ({zS }) = x ∈ Rd :
xi ≤ zS for all S .
(5.4.7)
i∈S
Assume that all the inequalities above are tight. Then, Arcila [11] showed that
P ({zS }) is a type B generalized permutahedron if and only if the values zS form a
bisubmodular function in the sense of Fujishige [42]. That is,
zS + zJ ≥ zS∩J + zS]J ,
where S ] J = (S ∪ J) \ (S ∪ J) and we set z∅ = 0. Moreover, every type B
generalized permutahedron can be written uniquely in this manner. See also [14].
Since zS = max heS , pi : p ∈ P ({zS }) , it follows that
P ({zS }) + λP ({zS0 }) = P ({zS + λzS0 })
for any λ ∈ R such that the (signed) Minkowski sum P ({zS })+λP ({zS0 }) is defined.
Proposition 5.4.11. For any {yS } ⊆ R such that the signed Minkowski sum
P
0
S yS ∆S is defined,
X
yS ∆0S = P ({zS }),
S
119
where zS =
X
yJ .
J : J∩S6=∅
Proof. The result immediately follows from the observation preceding the proposition and the following simple computation:
max heS , pi : p ∈ ∆0J =



1 if S ∩ J 6= ∅,


0 otherwise.
Theorem 5.4.12. Every type B generalized permutahedron can be written uniquely
as a signed Minkowski sum of the simplices {∆0S : S ⊆ [±d] involution-exclusive}.
Explicitly,
P ({zS }) =
X
yS ∆0S ,
(5.4.8)
S
X
where yS = (−1)d−|S|−1
(−1)|J| zJ .
J : J∩S=∅
Proof. To prove (5.4.8), it is enough to show that for all involution-exclusive S ⊆
[±d],
X
max heS , pi : p ∈
yK ∆0K = zS .
K
That is, that for all S
X
(−1)d−|K|−1
K : K∩S6=∅
X
(−1)|J| zJ = zS .
J : J∩K=∅
Regrouping terms, the previous sum is
X
J
d−|J|−1
(−1)
X
|K|
(−1)
zJ
K : K∩J=∅
K∩S6=∅
So we are only left to show that the internal sum is zero whenever J 6= S, and is
(−1)d−|S|−1 for J = S. We consider the different cases below. Recall that all sets
we consider are nonempty and involution-exclusive.
120
If there is j ∈ J \ S, we can pair sets K appearing in the sum in following
manner: K ↔ K∆{j}, where ∆ denotes the symmetric difference. Indeed, since
j ∈
/ J and j ∈
/ S, a set K appears in the sum if and only if K∆{j} does (the
condition K ∩S 6= ∅ guarantees K \{j} is nonempty). Since (−1)|K| +(−1)|K∆{j}| =
0, the overall sum is zero. Thus, from this point we assume J ⊆ S
Suppose there is s ∈ S \ J, and pick j ∈ J. We divide the sets K appearing in
the sum in two groups: those containing s and those not containing s. We pair the
sets in the first group via K ↔ K∆{j}. Since K ∩ J = ∅, no set contains j, thus
K∆{j} is involution exclusive and appears in the sum if and only if K does. Now,
we pair the sets in the second group (those not containing s) via K ↔ K∆{s}.
Since s ∈
/ J and s ∈
/ S, K∆{s} ∩ J 6= ∅ appears in the sum if and only if K ∩ J = ∅
does.
Finally, suppose J = S. Write any K appearing in the sum as A ∪ B, where
A = K ∩ S and B = K \ S. Then, the internal sum above becomes
X
∅6=A⊆S
(−1)
|A|
X
|B|
(−1)
.
B⊆[±d]\±S
By the binomial theorem, the first factor is (1 − 1)|S| − (−1)0 = −1. For the second
factor, choosing B in the sum corresponds to chose whether i ∈ B, i ∈ B, or
i, i ∈
/ B for each i ∈ [d] \ ±S. Hence, the second factor equals
X
(−1)|v| = (−1 + 1 − 1)d−|S| = (−1)d−|S|
v∈{−1,0,1}d−|S|
where |v| denotes the number of nonzero entries of v. Thus, identity (5.4.8) is
proved. Uniqueness follows from the number of elements in the generating set.
Proof of Theorem 5.4.9. Observe that there are exactly d zero-dimensional polytopes in this collection: ∆S with |S| = 1. Since they minimally generate all
121
translations in Rd and h1 (πdB ) = 3d − d − 1, the statement is equivalent to showing
that the collection
{log[∆S ] : S special, |S| ≥ 2} ∪ {log[∆0S ] : S special}
spans Ξ1 (πdB ).
(5.4.9)
By Theorem 5.4.12, the collection {log[∆0S ] : S involution-
exclusive} generates Ξ1 (πdB ), so it suffices to write each log[∆0S ] as a linear combination of the elements in (5.4.9). We do this inductively on |S|.
Since ∆0i is a translate of ∆0i , we have that log[∆0i ] = log[∆0i ] is in (5.4.9) for
all i ∈ [±d] (either i or i is in [d]).
We use λ = −1 and r = 1 in Theorem 5.1.6 to conclude that for any polytope
P:
log[−P ] = −
X
(−1)dim(Q) log[Q].
Q≤P
Applying this identity to P =
log[∆0S ] = −
∆0S
= −∆0S yields
X
X
(−1)J log[∆J ].
(−1)|J| log[∆0J ] +
J⊆S
J(S
The result now follows by induction.
5.5
Hopf monoid structure
Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra GP
in [1]. It contains many other interesting combinatorial Hopf monoids as submonoids. In this section we show that the valuation (5.1.1) and translation invariance (5.1.2) properties define a Hopf monoid quotient of GP.
122
5.5.1
The McMullen (co)ideal
As a species, GP[I] is the vector space with basis
GP[I] = {P ⊆ RI : P is a generalized permutahedron}.
The product µS,T is defined by
µS,T (P ⊗ Q) = P × Q,
for all permutahedra P ∈ GP[S] and Q ∈ GP[T ]. In particular, GP is a commutative monoid. Let F be the face of the braid arrangement in RI corresponding to the
composition (S, T ), and let v ∈ relint(F ). Then, for any P ∈ GP[I], the face Pv decomposes as a product of generalized permutahedra P |S × P/S , with P |S ∈ GP[S]
and P/S ∈ GP[T ], see [1, Proposition 5.2]. The coproduct is defined by
∆S,T (P ) = P |S ⊗ P/S .
Aguiar and Ardila also give the following grouping-free and cancellation-free formula for its antipode. For a generalized permutahedron P ∈ GP[I],
sI (P ) = (−1)|I|
X
(−1)dim(Q) Q.
(5.5.1)
Q≤P
We now introduce the subspecies Mc of GP. The space Mc[I] ⊆ GP[I] is the
subspace spanned by elements
P ∪Q+P ∩Q−P −Q
for P, Q ∈ GP[I] such that P ∪ Q is convex, (5.5.2)
and
P+t − P
for P ∈ GP[I] and t ∈ RI ,
(5.5.3)
where P+t denotes the Minkowski sum P +{t}. The sums and differences in (5.5.2)
and (5.5.3) correspond to the vector space structure of GP[I], not to Minkowski
sum or difference.
123
Remark 5.5.1. Recall from Section 5.1.2 that if P ∪ Q is a polytope, then P ∪ Q
and P ∩ Q are necessarily generalized permutahedra. Thus, the elements (5.5.2)
are indeed in GP[I].
The following result shows that Mc define relations compatible with the Hopf
monoid structure of GP. Ardila and Sanchez [15] prove a similar result for extended generalized permutahedra by realizing Mc as the kernel of a Hopf monoid
morphism.
Theorem 5.5.2. The subspecies Mc is an ideal and a coideal of GP. That is,
µS,T Mc[S]⊗GP[T ] ⊆ Mc[I] and
∆S,T Mc[I] ⊆ Mc[S]⊗GP[T ]+GP[S]⊗Mc[T ],
e defined by
for any I = S t T . Therefore, the quotient species Π
e = GP[I]/Mc[I]
Π[I]
inherits the Hopf monoid structure of GP.
Proof. For generators of Mc of the form (5.5.3), the result follows from the following
two observations. If P ∈ GP[S], Q ∈ GP[T ] and t ∈ RS , then
P+t × Q = (P × Q)+(t,0) .
If P ∈ GP[I] and t ∈ RT , then
∆S,T (P+t ) = (P |S )+tS ⊗ (P |S )+tT ,
where tS and tT denote the projections of t to RS and RT , respectively.
We will now focus on generators of Mc of the form (5.5.2). Fix an arbitrary
finite set I and a nontrivial decomposition I = S t T . Let v ∈ RI be any vector
in the interior of the corresponding face of the braid arrangement.
124
Suppose P, P 0 , P ∪ P 0 ∈ GP[S] and Q ∈ GP[T ]. Then,
(P ∪ P 0 ) × Q = (P × Q) ∪ (P 0 × Q),
(P ∩ P 0 ) × Q = (P × Q) ∩ (P 0 × Q),
and (P ∪ P 0 ) × Q = (P × Q) ∪ (P 0 × Q) is a polytope if and only if P ∪ P 0 is. It
follows that
µS,T (P ∪ P 0 + P ∩ P 0 − P − P 0 ) ⊗ Q
= (P × Q) ∪ (P 0 × Q) + (P × Q) ∩ (P 0 × Q) − P × Q − P 0 × Q ∈ Mc[I].
Since GP is commutative, this proves that Mc is an ideal.
Now, let P, Q, P ∪ Q ∈ GP[I]. There are two possibilities:
i.
The face (P ∪Q)v of P ∪Q is completely contained in P or in Q. Without loss
of generality, suppose the former. Then (P ∪Q)v = Pv and, necessarily, (P ∩Q)v =
Qv . Hence,
∆S,T P ∪ Q + P ∩ Q − P − Q = ∆S,T (P ) + ∆S,T (Q) − ∆S,T (P ) − ∆S,T (Q) = 0
ii.
The face (P ∪ Q)v is not contained in P nor in Q. Hence, (P ∪ Q)v =
Pv ∪ Qv and (P ∩ Q)v = Pv ∩ Qv . Expanding the first equality we have
(P ∪ Q)|S × (P ∪ Q)/S = (P |S × P/S ) ∪ (Q|S × Q/S ).
The union of two Cartesian products A×B and C ×D is again a Cartesian product
if and only if one contains the other or either A = C or B = D. By assumption,
the is no containment between Pv and Qv . We can therefore assume without loss
of generality that
P |S = Q|S .
(5.5.4)
Projecting to RS and RT , we further see that
(P ∪ Q)|S = P |S ∪ Q|S = P |S
and
125
(P ∪ Q)/S = P/S ∪ Q/S .
(5.5.5)
In particular, P/S ∪ Q/S is a generalized permutahedron. On the other hand,
expanding (P ∩ Q)v = Pv ∩ Qv , we have
(P ∩ Q)|S × (P ∩ Q)/S = (P |S × P/S ) ∩ (P |S × Q/S ) = P |S × (P/S ∩ Q/S ).
Comparing factors, we deduce
(P ∩ Q)|S = P |S
(P ∩ Q)/S = P/S ∩ Q/S .
and
(5.5.6)
Putting together (5.5.4), (5.5.5) and (5.5.6), we conclude
∆S,T P ∪ Q + P ∩ Q − P − Q = ∆S,T (P ∪ Q) + ∆S,T (Q ∩ Q) − ∆S,T (P ) − ∆S,T (Q)
= P |S ⊗ (P/S ∪ Q/S ) + P |S ⊗ (P/S ∩ Q/S ) − P |S ⊗ P/S − P |S ⊗ Q/S
= P |S ⊗ P/S ∪ Q/S + P/S ∩ Q/S − P/S − Q/S ∈ GP[S] ⊗ Mc[T ].
Thus, in either case we get ∆S,T P ∪ Q + P ∩ Q − P − Q ∈ Mc[S] ⊗ GP[T ] +
GP[S] ⊗ Mc[T ]. That is, Mc is a coideal of GP.
Comparing the generators of the (co)ideal Mc with the relations defining Mce
Mullen’s polytope algebra, it is natural to ask if Π[I]
agrees with Π(πI ), where
πI ⊆ RI is the standard permutahedron. The answer is no. For instance, in the
polytope algebra, the structure of R-vector space is defined so that
α
1
−1 =
α
− 1,
where the numbers over the segments denote their length. However, if α is an
irrational number, then
α
1
−α• −
α
−• ∈
/ Mc[I].
Nevertheless, the proof of the previous theorem works verbatim to show that the
Hopf monoid operations are well-defined in Π(πI ).
126
Theorem 5.5.3. The species Π defined by Π[I] = Π(πI ) is a Hopf monoid, with
product and coproduct defined for any decomposition I = S t T by
µS,T ([Q1 ] ⊗ [Q2 ]) = [Q1 × Q2 ]
and
∆S,T ([P ]) = [P |S ] ⊗ [P/S ]
for all classes of generalized permutahedra [Q1 ] ∈ Π[S], [Q2 ] ∈ Π[T ], and [P ] ∈
Π[I]. Moreover, Π is a Hopf monoid quotient of GP via the morphism P 7→ [P ].
It immediately follows from the definitions above that, if F ∈ Σ[Ad ] is the face
corresponding to a composition (S, T ) of [d], then
[P ] · HF = µS,T ◦ ∆S,T ([P ])
for all generalized permutahedra P ⊆ Rd . Thus, in the case of generalized permutahedra, the module structure of Section 5.3 is precisely the one induced from the
Hopf monoid structure above.
The antipode
The antipode formula of GP (5.5.1) descends to the quotient Π, but it is no longer
grouping-free. The Euler map (5.1.7) allows us to write the antipode formula of Π
in a very compact form:
sI ([P ]) = (−1)|I| [P ]∗ = (−1)|I| [−P ]−1 ,
where [−P ]−1 is the multiplicative inverse of the class [−P ] in the polytope algebra.
The last equality is [55, Theorem 12]. This surprising relation between two different
notions of inversion, one in McMullen’s polytope algebra and the other in Aguiar
and Ardila’s Hopf monoid, can be explained through the module structure of this
chapter.
127
Let τ ∈ RΣ[AI ] be the Takeuchi element of the braid arrangement in RI . It
is the Adams element of parameter −1. In particular, it is projective and the
corresponding Eulerian family {EX } is (5.3.5). For a generalized permutahedron
P , write
[P ] =
X
Pr,k
M
where Pr,k ∈
r,k
Ξr (πI ) · EX .
dim(X)=k
Then,
sI ([P ]) = [P ] · τ =
X
Pr,k · τ =
r,k
X
(−1)k Pr,k .
r,k
On the other hand, using Proposition 5.2.7 we have
([P ])∗ =
X
r,k
5.5.2
(Pr,k )∗ =
X
(−1)|I|−k Pr,k = (−1)|I| sI ([P ]).
r,k
Higher monoidal structures
We have just proved that Π is a Hopf monoid in the symmetric monoidal category (Sp, ·). The algebra str ucture of each space Π[I] defined by McMullen can
also be defined for GP. In both cases, this endows the species with the structure of a monoid in the symmetric monoidal category (Sp, ×) of species with the
Hadamard product. The Hadamard product of two species p and q is defined by
(p × q)[I] = p[I] ⊗ q[I].
Hence, a monoid in (Sp, ×) consists of a species p with an algebra structure on
each space p[I]. For generalized permutahedra, these structures are compatible in
a very special way.
Theorem 5.5.4. The species of generalized permutahedra GP and its quotient Π
are (2, 1)-monoids in the 3-monoidal category (Sp, ·, ×, ·).
128
See [6, Chapter 7] for the definition of higher monoidal categories and of
monoids in such categories. The notation (2, 1) indicates that GP is a monoid
with respect to the first two monoidal structures (Cartesian product and Minkowski
sum, respectively) and a comonoid with respect to the last (coproduct maps ∆S,T ).
Proof. We only discuss the remaining compatibility axioms: the compatibility
between Cartesian product and Minkowski sum, and the compatibility between
Minkowski sum and the coproduct.
The compatibility between Cartesian product and Minkowski sum boils down
to the identity
(P1 + P2 ) × (Q1 + Q2 ) = (P1 × Q1 ) + (P2 × Q2 )
for P1 , P2 ∈ GP[S] and Q1 , Q2 ∈ GP[T ], which one easily verifies for arbitrary sets
P1 , P2 ⊆ RS and Q1 , Q2 ⊆ RT .
On the other hand, the compatibility between Minkowski sum and the coproduct is equivalent to the following identity for generalized permutahedra P, Q ∈
GP[I]:
(P + Q)|S ⊗ (P + Q)/S = (P |S + Q|S ) ⊗ (P/S + Q/S ).
This follows by projecting the identity (P + Q)v = Pv + Qv to RS and RT , respectively, where v is any vector in the interior of the face of the braid arrangement
corresponding to the composition (S, T ).
The compatibility between Minkowski sum and the Hopf monoid operations
refines the last statement in Theorem 5.2.3; which, in the language of this section,
states that the maps µS,T ◦∆S,T are compatible with the Minkowski sum operation.
129
CHAPTER 6
TYPE B HOPF MONOIDS
The contents of this chapter are joint work with Aguiar [2]. The notions related
to sets with a fixed-point free involutions are essential to this chapter, and can be
reviewed in Section 2.2.4. We set one additional piece of notation.
Let I be a point with a fixed-point free involution, and X `B I be a type
B partition of I. Then, both X0 (the zero block of X) and X± := X \ {X0 }
(the collection of non-zero blocks of X) are sets with a fixed-point free involution,
the one induced by the involution of I. The following is an example of a type B
partition of [±4], where we omit the brackets of each individual block for simplicity
X = {2 4 2 4 , 1 3 , 1 3},
Thus, X0 = {2, 4, 2, 4} is a set of four elements, and X± = {1 3 , 1 3} is a set of two
blocks, each block containing two elements.
6.1
Type B species
A type B set species is a functor m from the category of finite sets with a fixedpoint free involution with involution-preserving bijections, to the category of sets.
Explicitly, a type B species m consists of:
1. For each finite set with a fixed-point free involution I, a set m[I] of mstructures.
2. For each bijection σ : I → J that preserves involutions (i.e. σ(i) = σ(i)), a
130
function m[σ] : m[I] → m[J ]. These functions satisfy
m[σ ◦ τ ] = m[σ] ◦ m[τ ]
and
m[Id] = Id .
(6.1.1)
In particular, they are bijections.
The involution of I is a (involution-preserving) bijection, and thus determines
a map m[I] → m[I]. Conditions (6.1.1) imply that this map is an involution on
m[I], which might or might not have fixed points. We denote the image of an
element x ∈ m[I] under this map by x. It follows that for all x ∈ m[I] and
involution-preserving bijection σ : I → J ,
m[σ](x) = m[σ](x).
(6.1.2)
A morphism f : m → n of type B species is a natural transformation of
functors. That is, f consists of a collection of maps fI : m[I] → n[I] such that
n[σ] ◦ fI = fJ ◦ m[σ] for all bijections σ : I → J that preserve involutions. It
follows from (6.1.2) that the involutions m[I] → m[I] in the previous paragraph
form an involution of type B species m → m.
We define type B analogs to some important species.
Example 6.1.1. The type B exponential species EB is the species with exactly
one structure ∗I on each finite set with a fixed-point free involution I.
Example 6.1.2. The species 1 and X are defined by






{∗∅ } if I = ∅,
I if |I| = 1,
1[I] =
and
X[I] =




∅
∅ otherwise.
otherwise,
131
They are the unit for the Cauchy product and the substitution product, respectively. See [6, Section 8.1.2]. Their type B analogs are






I
{∗∅ } if I = ∅,
B
B
and
X [I] =
1 [I] =




∅
∅
otherwise,
if |I| = 2,
otherwise.
Example 6.1.3. A type B linear order on I is a total order ` such that i ≤` j if
and only if j ≤` i. The species of type B linear orders is denoted by LB . Note
that the condition on ` ∈ LB [I] above implies that ` = rev(`), the reversal of the
linear order.
6.1.1
Type B generating functions
Let m be a type B species. The type B generating function m(x) of m is
m(x) =
X
md
d≥0
xd
,
(2d)!!
where md = |m[±d]|. This is analogous to the definition of exponential generating
function for a type A species, but with using the double factorial (2d)!! = 2d d!
instead of d!.
Example 6.1.4. The power series of the species 1B , XB and EB are respectively
1
2
X xd
= ex/2 .
(2d)!!
d≥0
x1
=x
2!!
132
6.2
6.2.1
Type B bimonoids
Type A species with an involution
Let SpA denote the category of pairs (p, θ) where p is a species and θ : p → p is
an involution in the category of species Sp. The morphisms in SpA are the same
as in Sp. In particular, they do not necessarily preserve involutions.
A pair (p, θ) in SpA is an involutive monoid if p is a usual monoid in Sp and
the following diagram commutes.
µ
p·p
p
βp,p
p·p
θ
θ·θ
p·p
µ
p
The morphism βp,p is the usual braiding of the category Sp, see [6, Chapter 8].
Namely, βp,p (x, y) = (y, x). Thus, a pair (p, θ) is an involutive monoid if p is a
monoid and
θI (x · y) = θT (y) · θS (x)
(6.2.1)
for all decompositions I = S t T , and structures x ∈ p[S] and y ∈ p[T ].
Observe that if θ is the trivial involution θ = id, then (p, id) is an involutive
monoid if and only if p is a commutative monoid. In more generality, observe
that if p is commutative, then (p, θ) is an involutive monoid if and only if θ is a
morphism of monoids.
Involutive comonoids are defined dually. Explicitly, an involutive comonoid
(p, θ) satisfies
θI (z)|S = θS (z/T )
and
133
θI (z)/S = θT (z|T ),
(6.2.2)
for all decompositions I = S t T , and structures z ∈ p[I]. An involutive bimonoid (resp. involutive Hopf monoid) is an involutive monoid and comonoid
(p, θ) such that p is a bimonoid (resp. Hopf monoid) in the usual sense.
Example 6.2.1. As observed above, any commutative monoid p yields an involutive monoid (p, id). Similarly, any cocommutative comonoid is an involutive
comonoid. In particular, the exponential species E and the species of partitions Π
give rise to involutive Hopf monoids (E, id) and (Π, id).
Example 6.2.2. Consider the species (L, rev) where rev is the reversal morphism:
i <rev(`) j if and only if j <` j. Then, (L, rev) is in an involutive Hopf monoid with
the usual operations of concatenation and restriction. Indeed, we clearly have
rev(` · `0 ) = rev(`0 ) · rev(`)
and
rev(`)|S = rev(`|S ) = rev(`/T ).
Similarly, the species of compositions Σ can be endowed with the involution rev,
which reverses the order of the blocks of a composition, to obtain an involutive
Hopf monoid.
To simplify the notation, we will sometimes drop the subscript I from the map
θI : p[I] → p[I], and will write θp for both the morphism of species p → p and each
of its components p[I] → p[I].
6.2.2
The action of Sp on SpB
We introduce an action of the category of species Sp on the category of type B
species SpB . Given a species p and a type B species m, we define a type B species
p · m on sets by
(p · m)[I] =
a
I=StT tS
134
p[S] × m[T ].
(6.2.3)
We decompose I into an involution-inclusive subset T and its complement, which
we further decompose into an involution-exclusive subset S and its image under
the involution. There is one such decomposition for each S ∈ P 0 (I), in particular
S and T are allowed to be empty. Thus, a (p · m)-structure on I is a tuple (x, y)
where x is a p-structure on an involution-exclusive subset S ∈ P 0 (I) and y is an
m-structure on I \ ±S.
For a involution preserving bijection σ : I → J , the map (p · m)[σ] sends the
component p[S]×m[T ] of (p·m)[I] to the component p[σ(S)]×m[σ(T )] of (p·m)[J ]
via the product map p[σ|S ] × m[σ|T ].
It is not hard to verify that there are natural isomorphisms
1 · m = m and (p · q) · m = p · (q · m).
They identify (∗∅ , x) ∈ (1 · m)[I] with x ∈ m[I], and ((x, y), z) ∈ ((p · q) · m)[I]
with (x, (y, z)) ∈ (p · (q · m))[I], respectively.
Proposition 6.2.3. The generating function of p · m is p(x)m(x).
Proof. Summing over k = |S| in (6.2.3), we have
(p · m)d =
d X
d
k=0
k
k
2 pk md−k = 2 d!
d
X
pk
k=0
This is precisely the coefficient of
xk
pk
k!
k≥0
d
!
X
xd
(2d)!!
k!
md−k
d−k
2 (d −
k)!
.
in
xk
mk
(2k)!!
k≥0
X
!
= p(x)m(x).
Example 6.2.4. Let TB be the species of transversals. That is, for every set with
a fixed-point free involution I,
TB [I] = {S ∈ P 0 (I) : 2|S| = |I|}.
135
Identifying (∗S , ∗∅ ) ∈ E · 1B [I] with S, we see that TB = E · 1B . Proposition 6.2.3
then yields TB (x) = ex , which recovers the fact that I has 2|I|/2 transversals.
Example 6.2.5. Let PB be the species of involution-exclusive subsets. That is,
for every set with a fixed-point free involution I, PB [I] = P 0 (I). Identifying
(∗S , ∗I\±S ) ∈ E · EB [I] with S, we see that PB = E · EB . Thus, PB (x) = e3x/2 and
we recover that I contains 3|I|/2 involution-exclusive subsets.
Example 6.2.6. Let OPB be the species of ordered involution-exclusive subsets.
That is, OPB [I] consists of all linear orders on involution-exclusive subsets of I.
For example, OPB [±2] consists of 13 elements: the empty order, 4 singleton orders
(1, 1̄, 2, 2̄), and 8 orders on transversals (12, 21, 1̄2, 21̄, 12̄, 2̄1, 1̄2̄, 2̄1̄). One
similarly verifies that OPB = L · EB . Thus,
OPB (x) =
ex/2
.
1−x
The number of OPB -structures on [±d] for the first values of d are 1, 3, 13, 79, 633.
This is the number of ways to sort a spreadsheet with d columns (OEIS: A010844).
Example 6.2.7. A type B linear order is completely determined by the order on
1
.
the first half of its elements. Thus, LB = L · 1B . It follows that LB (x) =
1−x
6.2.3
Type B objects, monoids, and comonoids
A type B object is an object of the category SpA ×SpB . That is, a type B object
is a triple (p, θ, m) where p is a species, θ : p → p is an involution of species, and m
is a type B species. A morphism (f, g) : (p, θp , m) → (q, θq , n) of type B objects
is a pair of morphism f : p → q and g : m → n in Sp and SpB , respectively. The
involution θ will play a central role in the definition of type B bimonoids below.
136
First, a type B monoid is a type B object (p, θ, m), such that (p, θ) is an
involutive monoid with product µ, together with a morphism of type B species
α : p · m → m making the following diagrams commute.
Id ·α
p·p·m
p·m
α
µ·Id
p·m
α
α
p·m
m
∼
=
ι·Id
m
1·m
Explicitly, the morphism α has components
αS,T : p[S] × m[T ] −→ m[I]
(x, y)
7−→ x · y
for each decomposition I = StT tS. The commutative diagrams above correspond
to the associativity and unitality axioms:
x · (y · z) = (x · y) · z
and
·w =w
for all decompositions I = R t S t T t S t R, structures x ∈ p[R], y ∈ p[S],
z ∈ m[T ], and w ∈ m[I]. Here, ∈ p[∅] denotes the unit of the monoid p as in
Section 2.3.1.
A morphism of type B monoids (f, g) : (p, θp , m) → (q, θq , n) is a morphism
of type B objects such that f : p → q is a morphism of monoids and the following
diagram commutes.
p·m
α
f
f ·g
q·n
m
α
n
That is, fI (x·y) = fS (x)·gT (y) for all decompositions I = S tT tS and structures
x ∈ p[S], y ∈ m[T ].
Type B comonoids and their morphisms are defined dually. The comodule
137
morphism δ : m → m · p of a type B comonoid (p, θ, m) has components
δS,T : m[I] −→ p[S] × m[T ]
z 7−→ (z|S , z/S )
for each decomposition I = S t T t S. This coaction satisfies the usual coassociativity
(z|RtS )|R = z|R
(z|RtS )/R = (z/R )|S
(z/R )/S = z/RtS
and counitality z|∅ = , z/∅ = z axioms. A morphism of comonoids (f, g) satisfies
gI (z)|S = fS (z|S ) and gI (z)/S = gT (z/S ) for all decompositions I = S t T t S and
structures z ∈ m[I].
Example 6.2.8. Consider the type B object of linear orders (L, rev, LB ), with
LB and (L, rev) as in Examples 6.1.3 and 6.2.2, respectively. We endow it with
the structure of a type B monoid and type B comonoid as follows. Given a decomposition I = S t T t S, a linear order ` ∈ L[S], and a type B linear order
`0 ∈ LB [J ],
αS,T (`, `0 ) = ` · `0
is the concatenation ` `0 rev(`)
For example, with S = {1, 3} ⊆ [±4] and T = {2, 2, 4, 4},
13 · 2442 = αS,T (13, 2442) = 13244231.
Now, for ` ∈ LB [I], define δS,T (`) = (`|S , `/S ) by
`|S
is the restriction of ` to S,
`/S
is the restriction of ` to T .
For example,
δ{1,2,4},{3,3} (13244231) = (24̄1̄, 33)
138
(Co)commutative (co)monoids
Let (p, θ, m) be a type B object. The involution θ of p induces an involution
0
of p · m as follows. The component θI0 maps p[S] × m[J ] to p[S] × m[J ]
θ0 = θp,m
via
θI0 (x, y) := (θS (x), y) = (θS (x), y),
where x = p[ ](x) ∈ p[S]. The equality θS (x) = θS (x) follows from the naturality
of the involution θ. To simplify notation, we will write x
e = θS (x).
A type B monoid (p, θ, m) is commutative if p is a commutative monoid and
the following diagram commutes.
α
p·m
θ0
m
α
p·m
That is, a monoid (p, θ, m) is commutative if p is commutative and for every
decomposition I = S t T t S, and structures x ∈ p[S], y ∈ m[T ],
x
e · y = x · y.
Cocommutative comonoids are defined dually.
(6.2.4)
Explicitly, a comonoid
(p, θ, m) is cocommutative if q is cocommutative and for every decomposition
I = S t T t S, and structure z ∈ m[I],
fS
z|S = z|
and
z/S = z/S .
Example 6.2.9. The type B (co)monoid of linear orders (L, rev, LB ) is cocommutative but not commutative. Commutativity fails in any nontrivial example, for
instance
12 · 33̄ = 1233̄2̄1̄,
but
e · 33̄ = 2̄1̄ · 33̄ = 2̄1̄33̄12.
12
139
Cocommutativity boils down to the fact that ` = rev(`). Explicitly,
fS .
`|S = rev(`)|S = rev(`)|S = rev(`|S ) = `|
The second cocommutativity condition follows trivially since, by definition, both
`/S and `/S are the restriction of the order ` to T .
6.2.4
Type B bimonoids
A type B object (p, θ, m) is a type B bimonoid if it is a type B monoid and
comonoid, p is a bimonoid in the usual sense, and the following diagram commutes
p·m
α
m
δ
p·m
µ(2) ·α
∆(2) ·δ
p·p·p·p·m
β0
p·p·p·p·m
where β 0 is the following composition of braiding maps and involutions:
0
β 0 = Id ·(βp,p · Id ◦ Id ·βp,p ◦ βp,p · Id) · Id ◦ Id · Id ·θp,p·m
The internal parenthesis interchanges the second and fourth factors, it corresponds
to one of the two possible paths in the usual hexagon diagram of a braided monoidal
category. That is, β 0 (x1 , x2 , x3 , x4 , y) = (x1 , x4 , xe3 , x2 , y). This is the type B compatibility axiom.
Explicitly, the type B compatibility axiom for a bimonoid (p, θ, m) requires that
for any two decompositions I = S t T t S = S 0 t T 0 t S 0 , and structures x ∈ p[S],
y ∈ m[T ],
:
(x · y)|S 0 = x|A · y|D · x/AtB
and
140
(x · y)/S 0 = (x/A )|B · y/D ,
(6.2.5)
where A = S ∩ S 0 , B = S ∩ T 0 , and D = T ∩ S 0 , as represented in the following
diagram.
'
$
'
$
'
S
T
&
S
:
S0
T0
S0
%
&
$
A
B
C
D
E
D
C
B
A
%
&
(6.2.6)
%
Observe that x/AtB = x
e|C ∈ p[C] and y/D ∈ m[E].
Example 6.2.10. We continue our example of linear orders by showing that
(L, rev, LB ) is a type B bimonoid. Let S, T , S 0 , T 0 and their intersections be as
above. For ` ∈ L[S] and `0 ∈ LB [T ], observe that in ` · `0 = ``0 rev(`)
ˆ elements of A (ordered according to `) precede elements of D (ordered ac-
cording to `0 ), which in turn precede elements of C (ordered according to
rev(`)), and
ˆ elements of B (ordered according to `) precede elements of E (ordered ac-
cording to `0 ).
Since the coaction morphism δ is given by restriction of linear orders, each component of the compatibility axiom (6.2.5) follows from one of the observations above.
:
In particular, note that rev(`)|C = rev(`|C ) = `/AtB .
Two immediate consequences of the compatibility axiom are the following.
δS,T ◦ µS,T (x, y) = (x, y)
(6.2.7)
δS,T ◦ µS,T (x, y) = (e
x, y) = θp,m (x, y).
(6.2.8)
In particular, the action maps µS,T are injective, while the coaction maps δS,T are
surjective.
141
A morphism of type B bimonoids (f, g) : (p, θp , m) → (q, θq , n) is simply
a morphism of both type B monoids and type B comonoids. These morphisms
satisfy two surprising properties.
First, the component f is completely determined by g, in the following sense.
Let S ∈ P 0 (I) and x ∈ p[S]. Then, for any y ∈ m[I \ ±S],
fS (x) = fS ((x · y)|S ) = gI (x · y)|S .
The first equality is (6.2.7) and the second is part of the definition of a morphism
of comonoids.
Second, the component f respects the involutions. Indeed, with the same setup
as above,
fS (e
x) = fS ((x · y)|S ) = gI ((x · y))|S = fS (x) · gT (y) |S = f]
S (x).
The first and last equalities are (6.2.8). Finally, naturality implies
fS (θp (x)) = fS (e
x) = f]
S (x) = θq (fS (x)).
6.3
Convolution modules and the antipode
In this section we work with (type B) species with values on a vector space. That
is, p[I] and m[I] are vector spaces over a fixed field k. The definition of p · m for
vector species is obtained by replacing Cartesian products and disjoint unions with
tensor products and direct sums, respectively.
142
6.3.1
Type A with an involution
Given two objects (p, θp ) and (q, θq ) of SpA , we define a new pair hom(p, q),
θ
by
hom(p, q)[I] = {f : p[I] → q[I]}
for all finite sets I, and
f θ = θq ◦ f ◦ θp ,
for all f ∈ hom(p, q)[I]. We emphasize that maps in hom(p, q)[I] are not required
to respect the involutions, otherwise we would have f θ = f .
We similarly define an involution on HomSpA (p, θp ), (q, θq ) = HomSp (p, q).
For f : p → q, define f θ : p → q to have components (f θ )I = (fI )θ for each finite
set I. The following lemmas show that the involution f 7→ f θ behaves well with
respect to the morphisms of monoids and comonoids.
Lemma 6.3.1. Let f : (p, θp ) → (q, θq ) be a morphism of (involutive) monoids,
then f θ is also a morphism of monoids.
Proof. This follows from the following simple computation:
fIθ (x · y) = θq fI (θp (x · y)) = θq fI (θp (y) · θp (x)) = θq fT (θp (y)) · fS (θp (x))
= θq fS (θp (x)) · θq fT (θp (y)) = fSθ (x) · fTθ (y).
Lemma 6.3.2. Let f : (p, θp ) → (q, θq ) be a morphism of (involutive) comonoids,
then f θ is also a morphism of comonoids.
Proof. We verify the claim directly:
fSθ (z|S ) = θq fS (θp (z|S )) = θq fS (θp (z)/T )
= θq fI (θp (z))/T = θq fI (θp (z)) |S = fIθ (z)|S .
143
A similar computation shows that fTθ (z/S ) = fIθ (z)/S , hence f θ is indeed a morphism of comonoids.
If p is a comonoid and q is a monoid, then hom(p, q) is naturally endowed with
a monoid structure as follows. Given f ∈ hom(p, q)[S] and g ∈ hom(p, q)[T ], the
product f · g is defined by
(f · g)(x) = f (x|S ) · g(x/S ),
for all x ∈ p[I]. See [7, Section 3.2] for details.
Moreover, if (p, θp ) is an involutive comonoid and (q, θq ) is an involutive monoid,
then the product above turns hom(p, q), θ into an involutive monoid. Indeed,
with f, g, and x as above:
(f · g)θ (x) = θq (f · g)(θp (x)) = θq f (θp (x)|S ) · g(θp (x)/S )
= θq f (θp (x/T )) · g(θp (x|T )) = θq g(θp (x|T )) · θq f (θp (x/T ))
(6.3.1)
= g θ (x|T ) · f θ (x/T ) = (g θ · f θ )(x).
We turn our attention back to HomSpA (p, θp ), (q, θq ) . The convolution of
f, g ∈ HomSpA (p, θp ), (q, θq ) = HomSp (p, q) is the following composition of morphisms of species:
f ∗ g := µ ◦ (f · g) ◦ ∆,
see [6, Section 1.2.4]. That is, for all x ∈ p[I],
(f ∗ g)I (x) =
X
fS (x|S ) · gT (x/S ).
I=StT
The identity element u ∈ HomSpA (p, θp ), (q, θq ) for the convolution product is
determined by u∅ (p ) = q and uI (x) = 0 whenever I 6= ∅.
A computation similar to (6.3.1) shows that for all f, g : (p, θp ) → (q, θq ),
(f ∗ g)θ = g θ ∗ f θ .
144
(6.3.2)
6.3.2
Type B
Let (p, θp , m) be a type B comonoid and (q, θq , n) be a type B monoid. We endow
HomSpA ×SpB (p, θp , m), (q, θq , n) with the structure of a left HomSp (p, q)-module.
Given h ∈ HomSp (p, q) and (f, g) ∈ HomSpA ×SpB (p, θp , m), (q, θq , n) , define
h ∗ (f, g) := (h ∗ f ∗ hθ , h ∗ g),
where the convolutions in the first component occur in HomSp (p, q), and
h ∗ g := α ◦ (h · g) ◦ δ.
That is, for all x ∈ m[I],
(h ∗ g)I (x) =
X
hS (x|S ) · gT (x/S ).
I=StT tS
The associativity in the first component follows from the usual associativity of
HomSp (p, q) and (6.3.2). In the second component, the associativity of ∗ is a
consequence of the coassociativity of (p, θp , m) and the associativity of (q, θq , n).
For regular species, if p is a bimonoid and q is a commutative monoid, then the
convolution of two monoid morphisms f, g : p → q is again a morphism of monoids.
A dual statement holds for comonoid morphisms, in this case p is required to be
cocommutative and q must be a bimonoid. The following are type B analogs of
these results.
Proposition 6.3.3. Suppose (p, θp , m) is a bimonoid and (q, θq , n) is a commutative monoid. If h : p → q and (f, g) : (p, θp , m) → (q, θq , n) are monoid morphisms,
then so is h ∗ (f, g).
Proof. The computation bellow uses the compatibility axiom of (p, θp , m), the
monoid morphism properties, and the commutativity of (q, θq , n), in that order.
145
Let I = S t T t S, x ∈ p[S] and y ∈ m[T ]. Then,
(h ∗ g)I (x · y) =
X
hS 0 (x · y)|S 0 · gT 0 (x · y)/S 0
S 0 ∈P 0 (I)
=
X
:
hS 0 x|A · y|D · x/AtB · gT 0 (x/A )|B · y/D
S 0 ∈P 0 (I)
=
X
:
hA (x|A ) · hD (y|D ) · hC (x/AtB ) · fB ((x/A )|B ) · gE (y/D )
S 0 ∈P 0 (I)
=
X
hA (x|A ) · hD (y|D ) · hθC (x/AtB ) · fB ((x/A )|B ) · gE (y/D )
S 0 ∈P 0 (I)
=
X
hA (x|A ) · fB ((x/A )|B ) · hθC (x/AtB ) · hD (y|D ) · gE (y/D )
S=AtBtC
D∈P 0 (T )
= (h ∗ f ∗ hθ )S (x) · (h ∗ g)T (y).
:
The step where we replace hC (x/AtB ) by hθC (x/AtB ) is an instance of (6.2.4).
A similar argument shows the following.
Proposition 6.3.4. Suppose (p, θp , m) is a cocommutative comonoid and (q, θq , n)
is a bimonoid. If h : p → q and (f, g) : (p, θp , m) → (q, θq , n) are morphisms of
comonoids, then so is h ∗ (f, g).
Proof. Let I = S 0 t T 0 t S 0 and z ∈ m[I]. Then,
(h ∗ g)I (z)|S 0 =
X
hS (z|S ) · gT (z/S ) |S 0
S∈P 0 (I)
=
X
:
hS (z|S )|A · gT (z/S )|D · hS (z|S )/AtB
S∈P 0 (I)
=
X
:
hA (z|A ) · fD (z|D ) · hC (z|C )
S∈P 0 (I)
=
X
θ
hA (z|A ) · fD (z|D ) · hC(z|
S∈P 0 (I)
= (h ∗ f ∗ hθ )(z|S 0 ).
146
C)
And
(h ∗ g)I (z)/S 0 =
X
hS (z|S ) · gT (z/S ) /S 0
S∈P 0 (I)
=
X
(hS (z|S )/A )|B · gT (z/S )/D
S∈P 0 (I)
=
X
hB ((z/S 0 )|B ) · gE ((z/S 0 )/B )
S∈P 0 (I)
= (h ∗ g)(z/S 0 ).
6.3.3
Type B Hopf monoids and the antipode
Recall that a bimonoid p is a Hopf monoid if Id ∈ EndSp (p) := HomSp (p, p) is
invertible with respect to the convolution product. In this case, the inverse of the
identity is denoted s and is called the antipode of p.
If (p, θ) is an involutive Hopf monoid, then Takeuchi’s formula (2.3.3) shows
that for all x ∈ p[I],
X
sI (θ(x)) =
(−1)k µS1 ,...,Sk ◦ ∆S1 ,...,Sk (θ(x))
(S1 ,...,Sk )I
=
X
(−1)k θ µSk ,...,S1 ◦ ∆Sk ,...,S1 (x) = θ(sI (x)).
(S1 ,...,Sk )I
That is, the antipode and the involution commute. In particular,
sθ = s
and
s ∗ Id ∗sθ = ∗ s = s.
A type B object (p, θ, m) is a type B Hopf monoid if it is a type B bimonoid
and p is a Hopf monoid. In this case, we define the type B antipode of (p, θ, m)
to be the pair
(s, s± ) := s ∗ (Idp , Idm ).
147
Explicitly, for x ∈ m[I],
X
s±
I (x) =
sS (x|S ) · x/S .
I=StT tS
Using Takeuchi’s formula (2.3.3) for the antipode of p and the (co)associativity of
(p, θ, m), we obtain the following type B analog of Takeuchi’s formula for s± :
X
s±
I (x) =
(S1 ,...,Sk ,T ,Sk ...,S1
(−1)k αS1 ,...,Sk ,T ◦ δS1 ,...,Sk ,T (x).
(6.3.3)
)B I
Observe that Idp ∗s± = Idp ∗s ∗ Idm = Idm . This yields the following recursive
formula to compute s± :
X
s±
I (x) = x −
x|S · s±
T (x/S ).
(6.3.4)
I=StT tS
S6=∅
We refer to this formula as the type B Milnor-Moore formula.
In general, the antipode s is not a morphism of Hopf monoids from p to itself.
Rather, it can be seen as a Hopf monoid morphism p → pop,cop (for details, see
[6, Section 1.2.9]). In concrete terms, this means that s reverses products and
coproducts:
sI (x · y) = sT (y) · sS (x)
and
∆S,T (sI (z)) = (sS ⊗ sT )(z/T ⊗ z|T ),
for all decompositions I = S t T and structures x ∈ p[S], y ∈ p[T ], z ∈ p[I]. The
following two propositions are the type B analogs of this result.
Proposition 6.3.5. The antipode (s, s± ) reverses products. That is, for all decompositions I = S t T t S, and structures x ∈ p[S], y ∈ m[T ],
s±
x) · s±
I (x · y) = sS (e
T (y).
148
(6.3.5)
Proof. We follow the notation in (6.2.6).
s±
I (x · y) =
X
sS 0 ((x · y)|S 0 ) · (x · y)/S 0
I=S 0 tT 0 tS 0
=
X
sS 0 (x|A · y|D · x
e|C ) · (x/A )|B · y/D
I=S 0 tT 0 tS 0
=
X
sC (e
x|C ) · sD (y|D ) · sA (x|A ) · (x/A )|B · y/D
I=S 0 tT 0 tS 0
If we first fix A t B = S \ (S ∩ S 0 ) and first sum over the possible values of
S ∩S 0 = A ⊆ AtB, the third and fourth factor are computing (s ∗Idp )AtB (x|AtB ).
By definition of the antipode, this is 0 unless A t B = ∅ (in which case C = S).
Thus, the sum above equals
X
sS (e
x) · sD (y|D ) · y/D = sS (e
x) · (s ∗ Idm )T (y) = sS (e
x ) · s±
T (y).
T =DtEtD
Proposition 6.3.6. The antipode (s, s± ) reverses coproducts. That is, for all
decompositions I = S 0 t T 0 t S 0 , and structures x ∈ m[I],
±
g
δS 0 ,T 0 (s±
I (x)) = sS 0 (x|S 0 ) ⊗ sT 0 (x/S 0 ).
Proof. The statement is clear if S 0 = ∅, so we assume that S 0 6= ∅ and consequently
T 0 is a proper subset. Applying δS 0 ,T 0 to (6.3.4) yields
δS 0 ,T 0 (s±
I (x)) = x|S 0 ⊗ x/S 0 −
X
δS 0 ,T 0 (x|S · s±
T (x/S )).
(6.3.6)
I=StT tS
S6=∅
By induction on the size of T , we have that for any decomposition T = D t E t D
:
±
δD,E (s±
T (x/S )) = sD ((x/S )|D ) ⊗ sE ((x/S )/D ),
hence, the sum above equals
X
: :
(x|S )|A · sD ((x/S )|D ) · (x|S )/AtB ⊗ ((x|S )/A )|B · s±
E ((x/S )/D ),
I=StT tS
S6=∅
149
(6.3.7)
with A, B, C, D, E as in (6.2.6). We first consider the terms where A = S 0 (thus,
C = D = ∅, B ∈ P 0 (T 0 ) can be chosen arbitrarily, and S = A t B). The sum of
these terms is
X
±
0
x|S 0 ⊗ (x/S 0 )|B · s±
E ((x/S 0 )/B ) = x|S 0 ⊗ (Idp ∗s )T 0 (x/S ) = x|S 0 ⊗ x/S 0 .
T 0 =BtEtB 0
We now consider the terms where A t B 6= ∅ and A ( S 0 (thus, C t D = S 0 \ A is
nonempty). If we first fix A and B satisfying these conditions, then all the factors
in the sum are constant except for the later two in the first component of the
tensor. Using the that θp reverses products (6.2.1), that it commutes with s, and
the coassociativity of p, we deduce
: :
sD ((x/S )|D ) · (x|S )/AtB = θp ((x|S )/AtB · sD ((x/S )|D ) = θp (y|C · sD (y/C )),
where θp (z) = θp (z) and y = (x/AtB )|S 0 \A . Since
X
θp (y|C · sD (y/C )) = θp ((Idp ∗s)S 0 \A (y)) = 0,
C⊆S 0 \A
the sum of the corresponding terms in (6.3.7) is zero. Finally, we consider the
terms where A = B = ∅ (thus, C t D = S 0 , C = S 6= ∅, and E = T 0 ). The sum of
these terms is
X
: :
sD ((x/C )|D ) · x|C ⊗ s±
T 0 ((x/C )/D )
D(S 0
=
X
:
:
:
±
sD ((x|S 0 )|D ) · (x|S 0 )/D ⊗ s±
T 0 (x/S 0 ) = −sS 0 ((x|S 0 )) ⊗ sT 0 (x/S 0 ).
D(S 0
: : :
We used coassociativity and (6.2.2) to deduce (x/C )|D = (x|S 0 )/C = (x|S 0 )|D , and
: :
similarly x|C = (x|S 0 )/D .
Finally, separating the sum (6.3.6) into the cases above, we obtain
:
:
= s ((x| )) ⊗ s± (x/
±
0
0
0
0
δS 0 ,T 0 (s±
(x))
=
x|
⊗
x/
−
x|
⊗
x/
+
0
−
s
((x|
)
)
⊗
s
(x/
)
S
S
S
S
I
S0
S0
S0
T0
S0
150
S0
T0
S 0 ).
If p is a Hopf monoid and q is a commutative monoid, then HomMon (p, q) is a
group under the convolution product; the inverse of f is f ◦ s. This statement has
two parts. First, it says that if f : p → q is a morphism of monoids, then so is
f ◦ s. Second, f ∗ (f ◦ s) = u, the unit of HomMon (p, q). The following is the type
B analog of this result.
Proposition 6.3.7. Let (p, θp , m) be a type B Hopf monoid with antipode (s, s± )
and (q, θq , n) be a type B commutative monoid. If (f, g) : (p, θp , m) → (q, θq , n) is a
morphism of monoids, then so is (f θ ◦ s, g◦ s± ). Moreover, f ∗(f θ ◦ s, g◦ s± ) = (f, g).
Proof. We verify the first statement directly. Let I = S t T t S, x ∈ p[S] and
y ∈ m[T ]. Then,
:
fSθ (sS (x)) · gT (s±
x)) · gT (s±
T (y)) = fS (sS (e
T (y))
= fS (sS (e
x)) · gT (s±
T (y))
hq is commutativei
= gI (sS (e
x) · s±
T (y))
h(f, g) is a morphism of monoidsi
= gI (s±
I (x · y))
h(s, s± ) reverses products (6.3.5)i
For the second statement, we use that f θ ◦ s is the convolution inverse of f θ
and consequently f θ ∗ (f θ ◦ s) ∗ (f θ )θ = f. Moreover, for all I and z ∈ m[I],
X
(f ∗ (g ◦ s± ))I (z) =
fS (z|S ) · gT (s±
T (z/S ))
I=StT tS
=
X
gI (z|S · s±
T (z/S )) = gI
I=StT tS
X
z|S · s±
T (z/S )
I=StT tS
= gI (Idp ∗s± )I (z) = gI (z).
Therefore, f ∗ (f θ ◦ s, g ◦ s± ) = (f, g).
151
6.4
Characters and polynomial invariants
A character of a Hopf monoid p is a morphism of monoids ζ : p → E. Since
the monoid E is commutative, the collection X(p) := HomMon (p, E) forms a group
under the convolution product called the group of characters of p, see [1, 4].
Fix a character ζ ∈ X(p). For every structure x ∈ p[I] we have a function
χI (x) : N → k defined by
χI (x)(n) := ζI∗n (x).
(6.4.1)
Expanding this definition, one obtains
χI (x)(n) =
X
ζI ◦ µS1 ,...,Sn ◦ ∆S1 ,...,Sn (x)
I=S1 t···tSn
=
|I| X
k=0
X
ζI ◦ µS1 ,...,Sk
I=S1 t···tSk
Si 6=∅
n
◦ ∆S1 ,...,Sk (x)
,
k
which shows that χI (x) is a polynomial of degree at most |I|. Moreover, the
naturality of ζ shows that χI (x) is an invariant of x; that is, χI (x) = χJ (y)
whenever y = p[σ](x) for some bijection σ : I → J.
Proposition 6.4.1 ([1]). The polynomial invariants χ satisfy:
1. For all decompositions I = S t T and structures x ∈ p[S] and y ∈ p[T ],
χI (x · y) = χS (x)χT (y).
2. For all structures x ∈ p[I] and scalars n, m,
χI (x)(n + m) =
X
χS (x|S )(n) χT (x/S )(m).
I=StT
3. For all structures x ∈ p[I] and scalars n,
χI (x)(−n) = χI (s(x))(n).
152
In this section we define characters of a type B Hopf monoid, define two polynomial invariants associated to a character, and prove properties analogous to those
of Proposition 6.4.1 for these new invariants.
6.4.1
Type A with an involution
Let (p, θp ) be an involutive Hopf monoid. Given a character ζ ∈ X(p), we define
the following two new polynomial invariants for each structure x ∈ p[I]:
ψI (x)(2n + 1) := ζ ∗(n+1) ∗ (ζ θ )∗n I (x),
ψI± (x)(2n) := ζ ∗n ∗ (ζ θ )∗n I (x).
(6.4.2)
(6.4.3)
To verify that ψI (x) and ψI± (x) are indeed polynomials of degree at most |I|, we
express them as a finite sum of polynomial functions of appropriate degrees, as
follows.
ψI (x)(2n + 1) =
X
χS (x|S )(n + 1)χθT (x/S )(n),
(6.4.4)
χS (x|S )(n)χθT (x/S )(n),
(6.4.5)
I=StT
ψI± (x)(2n) =
X
I=StT
where χθ denotes the polynomial invariants associated to the character ζ θ .
Proposition 6.4.2. The polynomial invariants ψ and ψ ± satisfy:
1. For all decompositions I = S t T and structures x ∈ p[S] and y ∈ p[T ],
ψI (x · y) = ψS (x)ψT (y)
and
ψI± (x · y) = ψS± (x)ψT± (y).
2. For all structures x ∈ p[I] and scalars t,
ψI (x)(−t) = (ψ θ )I (s(x))(t)
and
153
ψI± (x)(−t) = (ψ θ )±
I (s(x))(t).
Proof. Both ζ ∗(n+1) ∗ (ζ θ )∗n and ζ ∗n ∗ (ζ θ )∗n are characters, hence multiplicative,
and the first statement follows.
In order to verify the first claim in 2., it suffices to consider the case t = 2n + 1
where n is a positive integer. We use expression (6.4.4) and Proposition 6.4.1 to
deduce
ψI (x)(−2n − 1) =
X
χS (x|S )(−n) χθT (x/S )(−n − 1)
I=StT
=
X
χS (sS (x|S ))(n) χθT (sT (x/S ))(n + 1)
I=StT
=
X
χθT (sI (x)|T )(n + 1) χS (sI (x)/T ))(n)
I=StT
= (ψ θ )I (s(x))(2n + 1)
as claimed. A similar computation, but using (6.4.5) and t = 2n instead, proves
the claim for ψ ± .
6.4.2
Type B
Let (p, θp , m) be a type B Hopf monoid. A character of (p, θp , m) is a morphism of
type B monoids (ζ, ξ) : (p, θp , m) → (E, id, EB ), and X(p, θp , m) denotes the collection of characters of (p, θp , m). Extending the definition of the identity character
u ∈ X(p), we define (u, uB ) ∈ X(p, θp , m) by



1 if x = ∅ ,
B
uI (x) =


0 if I =
6 ∅.
Proposition 6.3.3 shows that the group X(p) acts on X(p, θp , m). Given a char-
154
acter (ζ, ξ) of (p, θp , m) and a positive integer n, we consider the characters:
ζ ∗n ∗ (ζ, ξ) = (ζ ∗(n+1) ∗ (ζ θ )∗n , ζ ∗n ∗ ξ)
(6.4.6)
ζ ∗n ∗ (u, uB ) = (ζ ∗n ∗ (ζ θ )∗n , ζ ∗n ∗ uB ),
(6.4.7)
and use them to define the following new polynomial invariants for each structure
x ∈ m[I]:
χI (x)(2n + 1) := ζ ∗n ∗ ξ I (x)
∗n
B
(x).
χ±
(x)(2n)
:=
ζ
∗
u
I
I
(6.4.8)
(6.4.9)
In order to verify that these invariants are indeed polynomial and of degree at
most |I|/2, we expand the definition
X
χI (x)(2n + 1) =
ξI ◦ αS1 ,...,Sn ,T ◦ δS1 ,...,Sn ,T (x)
I=S1 t···tSn tT tSn t···tS1
=
|I|/2 X
X
k=0
I=S1 t···tSk tT tSk t···tS1
Si 6=∅
ξI ◦ αS1 ,...,Sk ,T
n
◦ δS1 ,...,Sk ,T (x)
.
k
(6.4.10)
Observe that we have implicitly used that (ζ, ξ) is a character, and
ζS1 (x1 ) · . . . · ζSn (xn ) · ξT (y) = ξI (x1 · . . . · xn · y).
A similar expression with T = ∅ shows that χ±
I (x) is a polynomial invariant of
degree at most |I|/2.
Theorem 6.4.3. The polynomial invariants χ and χ± satisfy:
1. For all decompositions I = S t T t S and structures x ∈ p[S] and y ∈ m[T ],
χI (x · y) = ψS (x)χT (y)
and
±
±
χ±
I (x · y) = ψS (x)χT (y).
2. For all structures x ∈ p[I] and scalars s, t,
χI (x)(s + t) =
X
I=StT tS
155
χS (x|S )( 2s ) χT (x/S )(t),
and
χ±
I (x)(s + t) =
X
χS (x|S )( 2s ) χ±
T (x/S )(t).
I=StT tS
3. For all structures x ∈ p[I],
χI (x)(−1) = ξI (s±
I (x)).
Proof. The claims in 1. follow from the multiplicativity of the characters (6.4.6)
and (6.4.7), and the definition of the polynomial invariants involved.
To prove the first claim in 2., it is enough to consider the case s = 2n and
t = 2m + 1 for positive integers n, m. In this case, expanding definition (6.4.8)
yields
χI (x)(2(n + m) + 1) = ζ ∗(n+m) ∗ ξ I (x)
X
ζS∗n (x|S ) · ζ ∗m ∗ ξ T (x/S )
=
I=StT tS
=
X
χS (x|S )(n) · χT (x/S )(2m + 1),
I=StT tS
as we wanted to show. A similar analysis using definition (6.4.9) in the case s = 2n
and t = 2m with n, m positive integers yields the result for χ±
I (x).
For the last statement, we use (6.4.10) with n = −1 to deduce
χI (x)(−1) =
X
(S1 ,...,Sk ,T ,Sk ...,S1 )B I
−1
ξI ◦ αS1 ,...,Sk ,T ◦ δS1 ,...,Sk ,T (x).
k
The result follows by using the linearity of ξI and the type B analog of Takeuchi’s
formula (6.3.3).
156
6.5
Series
A series of a species p is a morphism s : E → p, see Section 2.3.2 for a more
explicit definition. Similarly, a series of a type B object (p, θp , m) is a morphism
(s, sB ) : (E, id, EB ) → (p, θp , m). Explicitly, a series of (p, θp , m) corresponds to a
choice of elements sI = sI (∗I ) ∈ p[I] and s0I = sB
I (∗I ) ∈ m[I] such that
p[σ](sI ) = sJ
m[τ ](s0I ) = s0J
and
for any bijection σ : I → J and any involution-preserving bijection τ : I → J .
In particular, the element s[n] is Sn -invariant and s[±n] is Bn -invariant. We let
S (p, θp , m) the space of series of (p, θp , m).
Let (p, θp , m) be a type B monoid. We consider the action of HomSp (E, p) on
HomSpA ×SpB (E, id, EB ), (p, θp , m) in Section 6.3.2. That is, we endow S (p, θp , m)
with the structure of a left module over the algebra S (p). The action of a series
t ∈ S (p) on a series (s, s0 ) ∈ S (p, θp , m) is determined by
t ∗ (s, s0 ) = (t ∗ s ∗ θ(t), t ∗ s0 ),
where the first component is the Cauchy product of the series t, s, θ(t) ∈ S (p),
the series θ(t) is defines by θ(t)I = θp (tI ) for all I, and
X
(t ∗ s0 )I =
tS · s0T .
I=StT tS
Example 6.5.1. We extend Example 2.3.4 to the type B case. A series (s, s0 ) of
(E, id, EB ) is determined by
s[n] = an ∗[n]
and
s0[±n] = bn ∗[±n]
for arbitrary scalars an , bn ∈ k. We identify (s, s0 ) with the following pair of formal
power series in k[[x]]
X
n≥0
an
xn
n!
and
X
n≥0
157
bn
xn
.
(2n)!!
Then, the action of S (E) ∼
= k[[x]] on (E, id, EB ) ∼
= k[[x]] × k[[x]] is given by
h(x) ∗ (f (x), g(x)) = (h2 (x)f (x), h(x)g(x)).
6.6
Substitution product of type B objects
The substitution product of species is defined in [6, Section 8.7.1]. If p is a species
and q is a positive species (i.e. q[∅] = ∅), the substitution product p ◦ q is the
species
(p ◦ q)[I] =
a
p[X] ×
Y
q[S] .
S∈X
X`I
Thus, a (p ◦ q)-structure on I is a pair (x, y) where x ∈ p[X] for some partition
X of I and y is a tuple of q-structures yS ∈ q[S], one for each block S ∈ X. The
species X is the unit with respect to the substitution product.
We define an analog of the substitution product for type B objects. We say
that a type B object (p, θp , m) is positive if p is a positive species. Observe that
we do not ask m[∅] to be empty.
Given two type B objects (p, θp , m) and (q, θq , n), the second one positive, define
a new type B object
(p, θp , m) ◦ (q, θq , n) := (p ◦ q, θp ◦ θq , m ◦B (q, n)),
where p ◦ q and θp ◦ θq come from the usual substitution product of species, and
(m ◦B (q, n))[I] :=
a
m[X± ] ×
Y
e
q[S] × n[X0 ].
S∈X±
X`B I
e
Q
q[S] denotes the e-invariant part of S∈X± q[S]. Namely, it conQ
fS .
tains tuples (xS )S∈X± ∈ S∈X± q[S] such that xS = x
Here,
Q
S∈X±
158
Remark 6.6.1. Related substitution operations of a (regular) species with a type
B (Hyperoctahedral) species are defined by Choquette in his Ph.D. thesis [35].
Example 6.6.2. Let us consider a (EB ◦B (E+ , EB ))-structure on I. It corresponds
to a triple (∗X± , (∗S )S∈X± , ∗X0 ), where {X0 } t X± is a type B partition of I. Observe that the triple is completely determined by the partition {X0 } t X± . Thus,
identifying (∗X± , (∗S )S∈X± , ∗X0 ) with {X0 } t X± , we deduce ΠB = EB ◦B (EB , E+ ).
Moreover, one easily verifies that
(Π, id, ΠB ) = (E, id, EB ) ◦ (E+ , id, EB ).
Proposition 6.6.3. The type B object (X, id, 1B ) is the unit of the substitution
product of type B objects.
Proof. Since X is the substitution unit in Sp, we only need to verify that
1B ◦B (q, n) = n
and
m ◦B (X, 1B ) = m.
For the first identity, observe that 1B [X± ] 6= ∅ if and only if X± = ∅, in
e
Q
which case
q[S]
consists of the empty tuple and X0 = I. Identifying
S∈X±
(∗∅ , (), x) ∈ (1B ◦B (q, n))[I] with x ∈ n[I], we deduce 1B ◦B (q, n) = n.
The second identity is deduced in a similar manner. Indeed, 1B [X0 ] 6= ∅ if
Q
and only if X0 = ∅ and S∈X± X[S] 6= ∅ if and only if each block S ∈ X± is a
singleton. The result follows by identifying (x, ({i})i∈I , ∗∅ ) ∈ (m ◦B (X, 1B ))[I]
with x ∈ m[I].
Proposition 6.6.4 (Type B compositional formula - species). Suppose q is a
positive species. The generating series of m ◦B (q, n) is m(q(x))n(x).
159
Proof. An application of the usual compositional formula [68, Theorem 5.1.4]
xd
in m(q(x))n(x) is
shows that the coefficient of
(2d)!!
(2d)!!
d X
1
k=0
k!
X
{S1 ,...,Sr }`[k]
=
mr
nd−k
q
.
.
.
q
|S1 |
|Sr |
r
d−k
2
2
(d − k)!
d X
d
k=0
k
X
2k−r mr q|S1 | . . . q|Sr | nd−k
{S1 ,...,Sr }`[k]
X
=
{S0 ,S1 ,...,Sr ,S1 ,...,Sr
mr q|S1 | . . . q|Sr | n|S0 |/2 ,
}`B [±d]
which is precisely the number of (m ◦B (q, n))-structures on [±d]. To verify the
last equality, note that choosing a type B partition {S0 , S1 , S1 , . . . , Sk , Sk } `B [±d]
with |S0 | = 2(d − k) is equivalent to:
1. choosing a subset S ∈
[d]
k
and setting S0 = ±([d] \ S),
2. choosing a partition {K1 , . . . , Kr } of S, and
3. constructing blocks {Si , Si } from Ki as follows: for each j ∈ Ki \ {max Ki },
choose whether j and max Ki will be in the same or in opposite blocks.
There are precisely 2k−r possible choices in the last step.
Example 6.6.5. Continuing Example 6.6.2, we deduce
B
B
B
Π (x) = E (E+ (x))E (x) =
ex −1 x
e 2 e2
=
ex +x−1
e 2 .
The number of ΠB -structures on [±d] for the first values of d are 1, 2, 6, 24, 116.
These are the type B Bell numbers, or Dowling numbers (OEIS: A007405).
Example 6.6.6. Some type B species can be obtained in more than one way using
the action of Sp on SpB and the substitution product. For instance, (Σ, rev, ΣB ) =
(L, rev, LB ) ◦B (E+ , id, EB ). Thus,
ΣB = E B · Σ
ΣB = LB ◦B (E+ , EB ).
and
160
The formulas in Proposition 6.2.3 and Proposition 6.6.4 give two “different” expressions for the same generating function:
ex/2 ·
1
1
· ex/2 .
=
x
2−e
1 − (ex − 1)
The number of ΣB -structures on [±d] for the first values of d are 1, 3, 17, 147. These
are the Type B Fubini numbers or Type B ordered Bell numbers (OEIS:
A080253)
Proposition 6.6.7. The substitution product on positive type B objects is associative.
m
r q
r
q
r
r
q
r
r
r
n
r
r
r
r
r q
r
r
o
r
r
m
r q
r
r
r
q
r
r
r
q
n
r
r
r
r
r q
r
r
o
r
r
Figure 6.6.1: Associativity of the type B substitution product.
Sketch of proof. The associativity in the first two components is the usual associativity of the substitution product in species. For the type B component, we need
161
to prove that
(m ◦B (q, n)) ◦B (r, o) = m ◦B (q ◦ r, n ◦B (r, o)).
By definition, the first species on a set I is
a a
Y
e
m[Y± ] ×
q[C] × n[Y0 ] ×
X`B I
Y`B X
while the second is
Y
a
m[W± ]×
W`B I
C∈Y±
±
a
q[V]×
B∈W± V`B
Y
Y
r[A]
e
× o[X0 ],
A∈X±
Y
e a
e
r[E]
×
n[Z± ]×
r[D] ×o[Z0 ] .
E∈V
Z`B W0
D∈Z±
Figure 6.6.1 represents how to take a structure of the first kind to a structure
of the second kind. Different colors represent different (type B) partitions. Solid
squares represent the blocks of a (type B) partition, a bold square represents
the zero block of a type B partition, and a dashed rounded rectangle represents
the collection of nonzero blocks of a type B partition. In the first picture, X is
represented in black and Y in red. In the second picture, W is green, the partitions
V are black and Z is orange.
A similar argument shows the following property of the substitution product.
Proposition 6.6.8. For all species p, q and type B species m, n,
(p · m) ◦B (q, n) = (p ◦ q) · (m ◦B (q, n)).
6.7
The type B bimonoids of set compositions and set partitions
We have already discussed how (Π, id) and (Σ, rev) yield involutive bimonoids.
In this section, we present how (Π, id, ΠB ) and (Σ, rev, ΣB ) can be endowed with
162
the structure of type B bimonoids. The (co)action maps naturally extend the
(co)multiplication of their type A component.
Fix a decomposition I = S t T t S. Given partitions X ` S, Y `B T , and
compositions F I, G B T , their products are defined by
X·Y
= X t Y t X,
F · G is the concatenation F G rev(F ).
It immediately follows that (Π, id, ΠB ) is commutative, while (Σ, rev, ΣB ) is not.
Observe that the type B partition X · Y satisfies
(X · Y)0 = Y0
and
(X · Y)± = X t Y± t X.
Conversely, given a type B partition X `B I, and a type B composition F =
(S1 , . . . , Sk , S0 , Sk , . . . , S1 ) B I, their coproduct is determined by
X|S = {S ∩ B : B ∈ X}b ,
X/S = {T ∩ B : B ∈ X}b ,
F |S = (S ∩ S1 , . . . , S ∩ Sk , S ∩ S0 , S ∩ Sk , . . . , S ∩ S1 )b ,
F/S = (T ∩ S1 , . . . , T ∩ Sk , T ∩ S0 , T ∩ Sk , . . . , T ∩ S1 )b .
where the superscript b denotes that all empty intersections (other than the zero
block in X/S `B T and F/S B T ) have been removed. Since X = X and
F = rev(F ), it follows that both (Π, id, ΠB ) and (Σ, rev, ΣB ) are cocommutative.
The compatibility axiom is verified following the argument in Example 6.2.10.
163
6.8
The free (commutative) monoid
The free monoid over a positive type B object (p, θ, m) is
T (p, θ, m) := (L, rev, LB ) ◦ (p, θ, m) = (L ◦ p, revθ , LB ◦B (p, m)).
A (L ◦ p)-structure on a finite set I is a tuple (F, p1 , . . . , pk ) where F = (S1 , . . . , Sk )
is a composition of I and pi ∈ p[Si ]. The involution revθ is determined by
revθ (F, p1 , . . . , pk ) = (rev(F ), θ(pk ), . . . , θ(p1 )),
where rev(F ) = (Sk , . . . , S1 ).
Similarly, a (LB ◦B (p, m))-structure on a set with an involution I is a tuple
(F, p1 , . . . , pk , m), where F = (S1 , . . . , Sk , T , Sk , . . . , Sk ) is a type B composition of
I, pi ∈ p[Si ], and m ∈ m[T ].
The type B object T (p, θ, m) is a type B monoid with the following product:
(F, p1 , . . . , pk ) · (G, p01 , . . . , p0r , m) = (F · G, p1 , . . . , pk , p01 , . . . , p0r , m).
(6.8.1)
The morphism (ι, ιB ) : (p, θ, m) → T (p, θ, m) of type B objects is determined by
ιI (p) = ((I), p)
ιB
I (m) = ((I), m)
and
(6.8.2)
for all p ∈ p[I] and m ∈ m[I]. Observe that the composition (I) of I does not
have nonzero blocks, so ((I), m) is indeed a (LB ◦B (p, m))-structure on I.
Proposition 6.8.1. Let (p, θp , m) be a positive type B object, (q, θq , n) be a type B
monoid, and (f, g) : (p, θp , m) → (q, θq , n) be a morphism of type B objects. Then
there is a unique morphism T (f, g) : T (p, θp , m) → (q, θq , n) of monoids that makes
the following diagram commute.
(p, θp , m)
(ι,ιB )
(f,g)
(q, θq , n)
T (f,g)
T (p, θp , m)
164
Proof. Define the morphism T (f ) : L ◦ p → q as in [6, Section 11.2]. That is,
T (f )I (F, p1 , . . . , pk ) = fS1 (p1 ) · · · · · fSk (pk ).
(6.8.3)
Observe that the expression on the right is a product of q-structures. Similarly,
define a morphism T 0 (f, g) : LB ◦B (p, m) → n by
T B (f, g)I (F, p1 , . . . , pk , m) = fS1 (p1 ) · · · · · fSk (pk ) · gT (m)
(6.8.4)
Define the morphism of type B objects T (f, g) := (T (f ), T B (f, g)). It immediately
follows from (6.8.1), (6.8.3), and (6.8.4) that T (f, g) is a morphism monoids, and
from (6.8.2) that the diagram above commutes. Uniqueness follows since every
(LB ◦B (p, m))-structure factors uniquely as product of structures of the form (6.8.2):
(F, p1 , . . . , pk , m) = ((S1 ), p1 ) · · · · · ((Sk ), pk ) · ((T ), m).
We can similarly define the free commutative monoid over a positive type
B object (p, θ, m):
S(p, θ, m) := (E, id, EB ) ◦ (p, θ, m) = (E ◦ p, θ0 , EB ◦B (p, m)).
A (E ◦ p)-structure on a finite set I is a pair (X, {pS }S∈X ), where X is a partition
of I and pS ∈ p[S] for all S ∈ X. The involution θ0 is determined by
θ0 (X, {pS }S∈X ) = (X, {θ(pS )}S∈X ).
Similarly, a (EB ◦B (p, m))-structure on I is a triple (X, {pS }S∈X± , m), where
X is a type B partition of I, the elements pS ∈ p[S] for all S ∈ X are such that
pS = pf
S , and m ∈ m[X0 ].
The type B object S(p, θ, m) is a type B (commutative) monoid with the following product:
(X, {pS }S∈X ) · (Y, {p0K }K∈Y± , m) = (X · Y, {pS }S∈X t {p0K }K∈Y± t {pS }S∈X , m)
165
The morphism (ι, ιB ) : (p, θ, m) → S(p, θ, m) of type B objects is determined by
ιI (p) = ({I}, {p})
ιB
I (m) = ({I}, ∅, m)
and
for all p ∈ p[I] and m ∈ m[I]. A proof similar to that of Proposition 6.8.1 shows
the following.
Proposition 6.8.2. Let (p, θp , m) be a positive type B object, (q, θq , n) be a commutative type B monoid, and (f, g) : (p, θp , m) → (q, θq , n) be a morphism of type
B objects. Then there is a unique morphism S(f, g) : S(p, θp , m) → (q, θq , n) of
monoids that makes the following diagram commute.
(p, θp , m)
(ι,ιB )
(f,g)
(q, θq , n)
S(f,g)
S(p, θp , m)
6.9
Type B Boolean functions
A Boolean function on a set I is a function z : 2I → R such that z(∅) = 0. Analogously, a type B Boolean function on a set with a fixed-point free involution
I is a function z : P 0 (I) −→ R such that z(∅) = 0. Let BFB denote the species
of type B Boolean functions and BF the usual species of Boolean functions. We
define an involution θ : BF → BF as follows. Let z ∈ BF[I], then for all A ⊆ I
θ(z)(A) := z(I \ A) − z(I).
Clearly θ(z)(∅) = 0, so θ(z) is again a Boolean function. Moreover, if z 0 = θ(z),
then
θ(z 0 )(A) = z 0 (I \ A) − z 0 (I) = z(A) − z(I) − z(∅) − z(I) = z(A),
so θ is indeed an involution.
166
Aguiar and Ardila [1] introduced a Hopf monoid structure on Boolean functions
that we review now. Fix a decomposition I = S t T . If z ∈ BF[S] and z 0 ∈ BF[T ],
their product is defined by
(z · z 0 )(A) := z(A ∩ S) + z 0 (A ∩ T )
for all A ⊆ I. Since BF is commutative, it follows that (BF, θ) is an involutive
monoid. On the other hand, the coproduct ∆S,T (z) of z ∈ BF[I] is determined by
z|S (A) := z(A)
z/S (B) := z(B ∪ S) − z(S),
and
for all A ⊆ S and B ⊆ T . We now verify that (BF, θ) is an involutive comonoid.
Indeed,
θ(z)|S (A) = θ(z)(A) = z(I \ A) − z(I)
= z(I \ A) − z(T ) − z(I) − z(T )
= z/T (S \ A) − z/T (S) = θ(z/T )(A),
for all A ⊆ S and
θ(z)/S (B) = θ(z)(B ∪ S) − θ(z)(S)
= z(I \ (B ∪ S)) − z(I) − z(I \ S) − z(I)
= z(T \ B) − z(T ) = z|T (T \ B) − z|T (T ) = θ(z|T )(B),
for all B ⊆ T .
We now endow (BF, θ, BFB ) with the structure of a commutative type B bimonoid. For a decomposition I = S t T t S and Boolean functions z ∈ BF[S],
z 0 ∈ BFB [T ], let αS,T (z, z 0 ) = z · z 0 be defined by
(z · z 0 )(A) := z(A ∩ S) + z 0 (A ∩ T ) + ze(A ∩ S)
167
(6.9.1)
for all A ∈ P 0 (I). Observe that ze = θ(z) is a Boolean function on S and z·z 0 = ze·z 0 .
Conversely, if z ∈ BFB [I], define δS,T (z) = (z|S , z/S ) by
z|S (A) := z(A)
z/S (B) := z(B ∪ S) − z(S)
and
for all A ⊆ S and B ∈ P 0 (T ). The (co)associativity of the (co)action can be
checked similarly to that of the (co)product of BF; we omit the details. We will,
however, verify the compatibility axiom.
0
Let I = S t T t S = S 0 t T 0 t S be two decompositions of I, and let their
pairwise intersections be as in (6.2.6). We need to verify that, for z ∈ BF[S] and
z 0 ∈ BFB [T ],
(z · z 0 )|S 0 = z|A · z|D · ze|C
and
(z · z 0 )/S 0 = (z/A )|B · z 0 /D .
The first condition readily follows from (6.9.1) since A = S 0 ∩ S, D = S 0 ∩ T , and
C = S 0 ∩ S. Now, for all K ⊆ T 0 ,
(z · z 0 )/S 0 (K) = (z · z 0 )(K ∪ S 0 ) − (z · z 0 )(S 0 )
= z((K ∪ S 0 ) ∩ S) + z 0 ((K ∪ S 0 ) ∩ T ) + ze((K ∪ S 0 ) ∩ S)
− z(S 0 ∩ S) + z 0 (S 0 ∩ T ) + ze(S 0 ∩ S)
= z((K ∩ B) ∪ A) − z(A) + z 0 ((K ∩ E) ∪ D) − z 0 (D) + ze((K ∩ B) ∪ C) − ze(C)
= z/A (K ∩ B) + z 0 /D (K ∩ E) + ze/C (K ∩ B) = ((z/A )|B · z 0 /D )(K),
as we wanted to show. The last equality uses that z/A (K ∩ B) = (z/A )|B (K ∩
:
:
B) and (e
z /C )|B = (z|A∪B )/A = (z/A )|B , which are instances of the involutive
comonoid axioms and the coassociativity of the coproduct.
168
6.10
Type B Submodular functions
Let SF denote the subspecies of BF consisting of submodular functions. That is,
Boolean functions z on I such that z(A) + z(B) ≥ z(A ∩ B) + z(A ∪ B) for all
A, B ⊆ I. The species SF is actually a sub Hopf monoid of BF, see [1]. The
involution θ of BF restricts to SF. Indeed,
θ(z)(A) + θ(z)(B) = z(I \ A) − z(I) + z(I \ B) − z(I)
≥ z((I \ A) ∪ (I \ B)) − z(I) + z((I \ A) ∩ (I \ B)) − z(I)
= z(I \ (A ∩ B)) − z(I) + z(I \ (A ∪ B)) − z(I)
= θ(z)(A ∩ B) + θ(z)(A ∪ B),
for all A, B ⊆ I. Thus, (SF, θ) is an involutive bimonoid.
Let SFB be the type B species of bisubmodular functions, in the sense of Fujishige [42]; see Section 5.4.3. Remember that z ∈ BF[I] is bisubmodular if
z(A) + z(B) ≥ z(A ∩ B) + z(A ] B)
(6.10.1)
for all A, B ∈ P 0 (I), where A ] B = (A ∪ B) \ (A ∪ B). The type B object
(SF, θ, SFB ) is a type B sub-bimonoid of (BF, θ, BFB ); that is, it is closed under the
action α and coaction δ.
Let I = S t T t S, z ∈ SF[S], and z 0 ∈ SFB [T ]. Observe that for A, B ∈ P 0 (I),
(A ∩ T ) ] (B ∩ T ) = (A ] B) ∩ T .
Thus, the bisubmodularity of z · z 0 follows from applying the submodularity of z
and ze, and the bisubmodularity of z 0 in (6.9.1).
Now, let z ∈ SFB [I] and I = S t T t S. The submodularity of z|S follows
directly from (6.10.1), since A ] B = A ∪ B for all A, B ⊆ S. The following simple
169
computation shows that z/S is bisubmodular. For all A, B ∈ P 0 (T ),
z/S (A) + z/S (B) = z(A ∪ S) − z(S) + z(B ∪ S) − z(S)
≥ z (A ∪ S) ∩ (B ∪ S) − z(S) + z (A ∪ S) ] (B ∪ S) − z(S)
= z (A ∩ B) ∪ S − z(S) + z (A ] B) ∪ S − z(S)
= z/S (A ∩ B) + z/S (A ] B).
6.11
Type B generalized permutahedra
Given a set with a fixed-point free involution I, let RI be the quotient of RI
where we have identified the basis element ei with −ei for all i ∈ I. For any
decomposition I = S t T t S, there is a canonical isomorphism RS × RT ∼
= RI
obtained by linearly extending (ei , 0) 7→ ei for all i ∈ S, and (0, ej ) 7→ ej for
all j ∈ T . In particular, we have RI ∼
= RI for any transversal I of I. In this
manner, we can define the type B Coxeter arrangement BI and type B generalized
permutahedra in RI just as in Section 2.2.4 and Section 5.4, respectively.
We endow (GP, θ, GPB ), where θ is the involution θ(P ) = −P , with the structure of a commutative type B Hopf monoid. The species GP is a Hopf monoid,
see [1] and Section 5.5. Moreover, the commutativity of GP and the identities
(−P ) × (−Q) = −(P × Q) and (−P )v = −(P−v ) show that (GP, θ) is an involutive
Hopf monoid. The action and coaction of GP on GPB are defined below.
Fix a decomposition I = S t T t S. For P ∈ GP[S] and Q ∈ GPB [T ], define
P · Q := P × Q ⊆ RS × RT ∼
= RI .
The normal fan of P coarsens (the fan consisting of the faces of) AS , the braid
arrangement in RS , and the normal fan of Q coarsens BT ; thus, the normal fan of
170
P × Q coarsens AS × BT , a subarrangement of BI . Hence, the polytope P · Q is a
type B generalized permutahedron. This product is clearly associative. Moreover,
P
P
observe that Pe = {x ∈ RS : −x ∈ P }, where
ai ei =
ai ei , is the image of P
under the canonical isomorphism RS ∼
= R±S ∼
= RS . Thus,
P × Q = Pe × Q,
and (GP, θ, GPB ) is commutative.
Let F ∈ Σ[BI ] be the face corresponding to the type B composition (S, T , S).
Then (BI )s(F ) = AS × BT . In particular, if P ⊆ RI is a type B generalized
permutahedron, then the normal fan of PF coarsens AS × BT . Consequently, there
are polynomials P |S ⊆ RS and P/S ⊆ RT such that
PF = P |S × P/S
and the normal fan of P |S (resp. P/S ) coarsens AS (resp. BT ). That is, P |S ∈
GP[S] and P/S ∈ GPB [T ]. Define the coaction by
δS,T (P ) = (P |S , P/S ).
We verify the compatibility axiom. Take P ∈ GP[S] and Q ∈ GPB [T ], and consider
a second decomposition I = S 0 t T 0 t S 0 . Then,
(P × Q)eS0 = PeS0 ∩S −eS0 ∩S × QeS0 ∩T = (P |A × (P/A )|B × P/AtB ) × (Q|D × Q/D ).
In the first equality, we use that the canonical isomorphism R±S ∼
= RS identifies
eS 0 ∩±S with eS 0 ∩S − eS 0 ∩S . The second equality uses the coassociativity of GP and
that eS 0 ∩S − eS 0 ∩S = (1)eA + (0)eB + (−1)eC , with A, B, C, D, E as in (6.2.6). The
0
result follows by taking the factors in RS and RT 0 in the expression above.
171
6.11.1
Isomorphism with (bi)submodular functions
Recall that there is a one-to-one correspondence between submodular functions
on I (resp. bisubmodular functions on I) and generalized permutahedra on RI
(resp. type B generalized permutahedra on RI ) [38] (resp. [11]). We show that
the morphism (SF, θ, SFB ) → (GP, θ, GPB ) arising from these correspondences is
an isomorphism of type B Hopf monoids.
Aguiar and Ardila [1] show that the component SF → GP is an isomorphism of
Hopf monoids, and one easily checks that it is in fact an isomorphism of involutive
Hopf monoids (SF, θ) → (GP, θ). Indeed, let z ∈ SF[I] be a submodular function
on I, and P (z) be the corresponding generalized permutahedron. That is,
P (z) = {x ∈ RI : x(I) = z(I) , x(S) ≤ z(S) for all S ⊆ I},
where x(S) =
X
xi . Then, −P (z) lies in the hyperplane x(I) = −z(I) = θ(z)(I),
i∈S
and inside this hyperplane is defined by inequalities −x(S) ≤ z(S) for all S.
Equivalently,
x(S) = x(I) − x(I \ S) ≤ −z(I) + z(I \ S) = θ(z)(S).
That is, −P (z) = P (θ(z)), as claimed.
Take a decomposition I = S t T t S and (bi)submodular functions z ∈ SF[S]
and z 0 ∈ SFB [T ]. We show that P (z) × P (z 0 ) = P (z · z 0 ), where P (z 0 ) and P (z · z 0 )
are defined as in (5.4.7). Indeed, for any x ∈ P (z) × P (z 0 ) and A ∈ P 0 (I),
x(A) = x(A ∩ S) + x(A ∩ T ) + x(S \ (A ∩ S)) − x(S)
≤ z(A ∩ S) + z 0 (A ∩ T ) + z(S \ (A ∩ S)) − x(S)
= z(A ∩ S) + z 0 (A ∩ T ) + θ(z)(A ∩ S).
172
In the last equality we use that x(S) = z(S). To check that this inequality is tight,
consider any point in the face P (z)F × Q(z 0 )G of P (z) × Q(z 0 ), where
F = (A ∩ S, S \ (±A ∩ S), A ∩ S) S
and
G = (A ∩ T , T \ (±A ∩ T ), A ∩ T ) B T .
Hence, P (z) × P (z 0 ) = P (z · z 0 ) and (SF, θ, SFB ) → (GP, θ, GPB ) is a morphism of
type B monoids.
Now, take a decomposition I = S t T t S and a bisubmodular function z ∈
SFB [I]. To show that P (z)|S = P (z|S ) ⊆ RS and P (z)/S = P (z/S ) ⊆ RT , we
verify that
P (z)|S × P (z)/S = P (z) ∩ {x ∈ RI : x(S) = z(S)}
and P (z|S ) × P (z/S ) are equal.
(⊆) Let x ∈ P (z) ∩ {x ∈ RI : x(S) = z(S)}. Since x ∈ P (z), for all S ⊆ S we
have x(A) ≤ z(A) = z|S (A). Moreover, for all B ∈ P 0 (T ) we have
x(B) = x(B t S) − x(S) = x(B t S) − z(S) ≤ z(B t S) − z(S) = z/S (B).
Thus, x ∈ P (z|S ) × P (z/S ).
(⊇) Let x ∈ P (z|S )×P (z/S ). First observe that x(S) = z|S (S) = z(S), so we only
need to show that x ∈ P (z). First observe that using bisubmodularity (6.10.1)
twice, we obtain
z(C) ≥ z(C ∩ S) + z(C ] S) − z(S)
hA = C, B = Si
≥ z(C ∩ S) + z((C ∩ T ) t S) + z(S \ (C ∩ S)) − 2z(S) hA = C ] S, B = Si
= z|S (C ∩ S) + z/S (C ∩ T ) + z(S \ (C ∩ S)) − z(S)
173
for all C ∈ P 0 (I). Thus,
x(C) = x(C ∩ S) + x(C ∩ T ) + x(C ∩ S)
= x(C ∩ S) + x(C ∩ T ) + x(S \ (C ∩ S)) − x(S)
≤ z|S (C ∩ S) + z/S (C ∩ T ) + z|S (S \ (C ∩ S)) − x(S) ≤ z(C),
implying that x ∈ P (z). The first inequality uses the definition of P (z|S ) and
P (z/S ), and the second uses that x(S) = z(S) and the previous equation.
Therefore, (SF, θ, SFB ) and (GP, θ, GPB ) are isomorphic type B Hopf monoids.
6.12
Symplectic matroids
A symplectic matroid on a set with a fixed-point free involution I is a collection
of involution-exclusive subsets M ⊆ P 0 (I) that is closed under inclusions and
satisfies the following axiom.
For all X, Y ∈ M such that |X| < |Y |, either
1. there is y ∈ Y \ X such that X ∪ {y} ∈ M , or
2. X ∪ Y is inadmissible and there is z ∈
/ X ∪ Y such that X ∪ {z} ∈ M
and {z} ∪ X \ Y ∈ M .
The elements of M are the independent sets of the matroid. A maximal independent set is a basis of M . Symplectic matroids were originally introduced by
Gelfand and Serganova [43, 44]. The characterization in terms of independent sets
above is originally due to Chow [36]. We let MB denote the type B species of
symplectic matroids.
174
Let M denote the Hopf monoid of Matroids. A matroid M ∈ m[I] is a nonempty
collection of subsets of I that is closed under inclusions and satisfies: whenever
X, Y ∈ M with |X| < |Y |, there is y ∈ Y \ X such that X ∪ {y} ∈ M . Sets in M
are the independent sets of the matroid M . A maximal subset in M is called a
basis of M . We recall the Hopf monoid structure of M below, see [1] for details.
Given a decomposition I = S t T , the product of matroids M1 ∈ M[S] and
M2 ∈ M[T ] is
M1 · M2 = {X ⊆ I : X ∩ S ∈ M1 , X ∩ T ∈ M2 }.
This is the direct sum of M1 and M2 . Conversely, for a matroid M ∈ M[I], the
restriction of M to S, sometimes called the deletion of T from M , is the matroid
M |S = M \ T = {X ⊆ S : X ∈ M } ∈ M[S],
and the contraction of S from M is
M/S = {X ⊆ T : X t B ∈ M for any basis B of M |S } ∈ M[T ].
The species of matroids M has a natural involution given by duality: M 7→ M ∗
for any M ∈ M[I]. Since M is commutative and
(M |S )∗ = (M \ T )∗ = M ∗ /T
we have that (M,
∗
and
(M/S )∗ = M ∗ \ S = M ∗ |T ,
) is an involutive Hopf monoid.
We proceed to endow (M,
∗
, MB ) with the structure of a commutative type B
Hopf monoid. Let I = S t T t S be a decomposition of I and M ∈ MB [I]. The
restriction of M to S is the matroid
M |S := {X ⊆ S : X ∈ M }.
175
Since subsets of S are always admissible, Chow’s axiom above implies the augmentation property for M |S , so M |S is indeed a matroid on ground set S. On the
other hand, the contraction of S form M is the symplectic matroid
M/S := {X ∈ P 0 (T ) : X t B ∈ M for any basis B of M |S }.
Lemma 6.12.1. The collection M/S above is well-defined and is a symplectic
matroid on ground set T .
Proof. We first show that the definition of M/S is independent of B. Let B1 , B2 be
bases of M |S and suppose X ∈ P 0 (T ) is such that X t B1 ∈ M but X t B2 ∈
/ M.
Choose a maximal subset Y ⊆ X such that Y t B2 ∈ M . Since |X t B1 | > |Y t B2 |
and (X t B1 ) ∪ (Y t B2 ) ⊆ X t K is admissible, Chow’s axiom implies there exists
z ∈ (X t B1 ) \ (Y t B2 ) such that (Y t B2 ) ∪ {z} ∈ I. Since B2 is a basis of M |S ,
the element z cannot be in S, so necessarily z ∈ X. However, this contradicts
the maximality of Y ⊆ X. Therefore, M/S is well-defined and independent of the
choice of basis B of M |S .
Now, fix a basis B ∈ M |S . It is clear that M/S = {X ∈ P 0 (T ) : X tB ∈ M } is
subset-closed, so we are only left to verify Chow’s axiom for M/S . Let X, Y ∈ M/S
with |X| < |Y | and suppose there is no z ∈ Y \ X such that X ∪ {z} ∈ M/S .
Then X t B, Y t B ∈ M are such that |X t B| < |Y t B| and no z ∈ (Y t B) \
(X t B) satisfies (X t B) ∪ {z} ∈ M . Therefore, Chow’s axiom for M implies
that (X t B) ∪ (Y t B) is inadmissible and there is z ∈
/ (X t B) ∪ (Y t B) such
that (X t B) ∪ {z} ∈ M and {z} ∪ (X t B) \ (Y t B) = {z} ∪ (X \ Y ) t B ∈ M .
Since B is a basis of M |S , this implies z, z ∈
/ S. That is, z ∈ T \ (X ∪ Y ) is such
that X ∪ {z} ∈ M/S and {z} ∪ (X \ Y ) ∈ M/S . Therefore, M/S is a symplectic
matroid.
176
We define the coproduct of (M,
∗
, MB ) by δS,T (M ) = (M |S , M/S ). The coas-
sociativity axiom can be easily verified from the definitions.
Before introducing the product of (M,
∗
, MB ), we recall the following charac-
terization of symplectic matroids due to Gelfand and Serganova.
Theorem 6.12.2 ([29, Theorem 3.3.3]). Let B be a collection of admissible subsets
of I of the same cardinality. Then, B is the collection of bases of a symplectic
matroid if and only if
PB := Conv{eB : B ∈ B}
if a type B generalized permutahedron.
Remark 6.12.3. The reader might be familiar with a similar result for usual matroids and generalized permutahedra. Observe that in type A, we might drop the
hypothesis that the sets in B have the same cardinality, since otherwise PB cannot
possibly be a generalized permutahedron. In type B, however, there are collections of admissible subsets of I of different cardinalities such that the convex hull
of their indicator vectors is a type B generalized permutahedron. These sets, of
course, cannot be the collection of basis of a symplectic matroid.
Let I = S t T t S be a decomposition of I, and M ∈ M[S] and M 0 ∈ MB [T ].
We define the product M · M 0 to be the symplectic matroid
f}
M · M 0 := {X ∈ P 0 (I) : X ∩ S ∈ M, X ∩ T ∈ M 0 , X ∩ S ∈ M
Let B be the collection of maximal elements of M · M 0 . Then,
B = {B t B 0 t (S \ B) : B is a basis of M and B 0 is a basis of M 0 }.
Observe that the elements in B have the same cardinality, and that
PB = (2PM − eS ) × PM 0 ,
177
where we write PM ⊆ RS and PM 0 ⊆ RT for the basis polytopes of M and M 0
respectively. Since PM is a generalized permutahedron and PM 0 is a type B generalized permutahedron, it follows from Theorem 6.12.2 that M · M 0 is indeed a
symplectic matroid. The associativity and commutativity of the product are clear
from the definitions.
Finally, we verify the compatibility axiom (6.2.5). Take two decompositions
I = S t T t S = S 0 t T 0 t S 0 of I, and let M ∈ M[S] and M 0 ∈ M[T ]. Define sets
A, B, C, D, E as in (6.2.6). The compatibility axiom requires us to verify that
:
(M · M 0 )|S 0 = M |A · M 0 |D · M/AtB
and
(M · M 0 )/S 0 = (M/A )|B · M 0 /D .
We do so explicitly. For the first identity, observe that
f}
(M · M 0 )|S 0 = {X ⊆ S 0 : X ∩ S ∈ M, X ∩ T ∈ M 0 , X ∩ S ∈ M
f}
= {X ⊆ S 0 : X ∩ A ∈ M, X ∩ D ∈ M 0 , X ∩ C ∈ M
f| },
= {X ⊆ S 0 : X ∩ A ∈ M |A , X ∩ D ∈ M 0 |D , X ∩ C ∈ M
C
:
f| . Similarly, fix a basis
the result follows since M/AtB = (M/AtB )∗ = M ∗ |C = M
C
B0 of (M · M 0 )|S 0 , then
(M · M 0 )/S 0 = {X ∈ P 0 (T 0 ) : X t B0 ∈ M · M 0 }
f}.
= {X ∈ P 0 (T 0 ) : (X t B0 ) ∩ S ∈ M, (X t B0 ) ∩ T ∈ M 0 , (X t B0 ) ∩ S ∈ M
Since X ⊆ T 0 and B0 ⊆ S 0 , we have that (X t B0 ) ∩ S = (X ∩ B) t BA , where
BA = B ∩ A is a basis of M |A . Thus, condition (X t B0 ) ∩ S ∈ M is equivalent to
X ∩ B ∈ (M/A )|B . A similar analysis for the other two conditions shows
f/ )| }.
(M ·M 0 )/S 0 = {X ∈ P 0 (T 0 ) : X∩B ∈ (M/A )|B , X∩E ∈ M 0 /D , X∩B ∈ (M
C B
The result follows by the definition of the product in (M,
: :
^
f/ )| = (M
(M
|AtB )|B = (M |AtB )/A = (M/A )|B .
C B
178
∗
, MB ) and because
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